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Bivariate symmetry tests with censored data

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Title:
Bivariate symmetry tests with censored data
Creator:
Perkins, Laura Lynn, 1957-
Copyright Date:
1984
Language:
English

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Laura Lynn Perkins. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
11720767 ( OCLC )
ACP2438 ( LTUF )
0030514176 ( ALEPH )

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Full Text
BIVARIATE SYMMETRY TESTS WITH CENSORED DATA

B Y
LAURA LYNN PERKINS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

1984




BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
BY
LAURA LYNN PERKINS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1984


to my parents, with
love


ACKNOWLEDGEMENTS
I would like to thank Dr. Ronald Randles for originally
proposing the problem. Without his enormous patience,
encouragement and guidence, it would not have been
possible. I would also like to thank Dr. Jim Kepner for his
help in its original conception. To my family, especially
my parents, I am grateful for the mental and financial
support they provided when I needed it the most. I would
like to thank Robert Bell for his patience and
understanding. More than once, when I could not see the
end, he was there to reassure me and give me confidence. To
my typist, Brenda Prine, I express my gratitude for many
hours spent with no complaints. Last, but not least, I
would like to say thank you to the Department of Statistics
for making this all possible.
iii


TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iii
ABSTRACT vi
CHAPTER
ONE INTRODUCTION 1
TWO A STATISTIC FOR TESTING FOR DIFFERENCES
IN SCALE 16
2.1 Introduction 16
2.2 The CD Statistic 19
2.3 Permutation Test 35
2.4 Asymptotic Results 39
2 .5 Comments 44
THREE A CLASS OF STATISTICS FOR TESTING FOR
DIFFERENCES IN SCALE 48
3.1 Introduction 48
3.2 u 2 Known 50
3.3 p^ = P 2 Unknown 62
3.4 Asymptotic Properties 73
3 .5 Comment s 86
FOUR A TEST FOR BIVARIATE SYMMETRY VERSUS
LOCATION/SCALE ALTERNATIVES 90
4.1 Introduction 90
4.2 The W Statistic Using
4.3 The Wn Statistic Using
CD Ill
4.4Permutation Test 121
4.5Estimating the Covariance 123
IV


FIVE MONTE CARLO RESULTS AND CONCLUSION. . 133
5.1 Introduction 133
5.2 Monte Carlo for the Scale Test . 134
5.3 Monte Carlo for the Location/
Scale Test 142
APPENDICES
1 TABLES OF CRITICAL VALUES FOR TESTING
FOR DIFFERENCES IN SCALE 158
2 THE MONTE CARLO PROGRAM 171
BIBLIOGRAPHY 183
BIOGRAPHICAL SKETCH 185
v


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
By
Laura Lynn Perkins
Augus t, 19 8 4
Chairman: Dr. Ronald H. Randles
Major Department: Statistics
Statistics are proposed for testing the null hypothesis
of bivariate symmetry with censored matched pairs. The two
types of alternatives considered are (1) the marginal
/
distributions have a common location parameter (either known
or unknown) and differ only in their scale parameters and
(2) the marginal distributions differ in their locations
and/or scales. For the first alternative, two types of
statistics are proposed. The first is a statistic based on
Kendall's tau modified for censored data, while the second
type is a class of statistics consisting of linear
combinations of two statistics. Conditional on N^, the
number of pairs in which both members are uncensored, and
N2 the number of pairs in which exactly one member is
censored, the two statistics used in the linear combination
are independent and each has a null distribution equivalent
vi


to that of a Wilcoxon signed rank statistic. Thus, any
member in the class can be used to provide an exact test
which is distribution-free for the null hypothesis. The
statistic based on Kendall's tau is not distribution-free
for small sample sizes and thus, a permutation test based on
the statistic is recommended in these cases. For large
samples, a modified version of the Kendall's tau statistic
is shown to be asymptotically distri bution-free.
For the second and more general alternative, a small
sample permutation test is proposed based on the quadratic
form Wn = T^ j; ^ T where T' is a 2-vector of statistics
composed of a statistic designed to detect location
differences and a statistic designed to detect scale
differences and | is the variance-covariance matrix for
T For large samples, a distribution-free approximation
for T' 1 T is recommended.
~ n T ~ n
Monte Carlo results are presented which compare the two
types of statistics for detecting alternative (1), for
sample sizes of 25 and 40. Quadratic form statistics Wn
using different scale statistic components are also compared
in a simulation study for samples of size 35. For the
alternative involving scale differences only, the statistic
based on Kendall's tau performed best overall but requires a
computer to do the calculations for moderate sample sizes.
For the more general alternative of location and/or scale
differences, the quadratic form using the scale statistic
based on Kendall's tau performed the best overall.
vii


CHAPTER ONE
INTRODUCTION
Let Wj and denote random variables; then the
property of bivariate symmetry can be defined as the
property such that (W ^ W 2 ) has the same distribution as
(W2*W^). This property of bivariate symmetry is also
referred to as exchangeability (or bivariate
exchangeability). Commonly, this property arises as the
null hypothesis in settings in which a researcher has paired
observations, such as, when the subjects or sampling units
function both as the treatment group and the control group
or possibly the researcher has matched the subjects
according to some criteria such as age and sex.
For example, a dentist may want to assess the
effectiveness of a dentifrice in reducing dental
sensitivity. The dentist randomly selects n patients and
schedules two appointments for each patient at three month
intervals. During the first visit, a hygienist assesses the
patient's dental sensitivity after which the patient is
given the dentifrice by the dentist. At the end of the
three month usage period, the patient returns and his or her
f
dental sensitivity is again assessed. If X^ and X2^ are
the first and second sensitivity measurements, respectively,
1


2
t" Vi
of the i 1 patient, the dentist has n bivariate pairs in the
sample. If there is no treatment effect, then effectively
the two observations of dental sensitivity are two
measurements of exactly the same characteristic at two
randomly chosen points of time. In which case, the
ft t f
distribution of is t^ie same as that of (X2i,X^i),
and so a test using the null hypothesis of bivariate
symmetry would be appropriate.
The possible alternatives for a test which uses a null
hypothesis of bivariate symmetry are numerous. The three
types of alternatives which will be considered in this work
are the following:
1) The marginal distributions have a common
known location parameter and differ only in
their scale parameters.
2) The marginal distributions have a common
unknown location parameter and differ only in
their scale parameters.
3) The marginal distributions differ in their
location and/or scale parameters.
The situation under consideration in this work is
further complicated by the possibility of censoring.
Censoring occurs whenever the measurement of interest is not


3
observable due to a variety of possible reasons. The most
common situation is when the measurement is the time to
"failure" (i.e., death, the time until a drug becomes
effective, the length of time a drug remains effective,
etc.) for an experimental unit subjected to a specific
treatment. If at the end of the experiment, the
experimental unit still has not "failed," then the
corresponding time to "failure" (referred to as survival
time) is censored. All that is known, is that the survival
time is longer than the observation time for that unit and
thus has been right censored. An example of censoring in
bivariate pairs could be the times to failure of the left
and right kidneys or the times to cancer detection in the
left and right breasts (Miller, 1981).
Many different types of right censoring exist (Type I,
Type II and random right censoring), each determined by
restrictions placed on the experiment. Type I censoring
occurs if the observation time for each experimental unit is
preassigned some fixed length T. Thus, if the survival time
for a unit is larger than T, it is right censored. Type II
censoring occurs when the experiment is designed to be
terminated as soon as the r*"^1 (r occurs. Random right censoring is a generalization of Type
I censoring, in which the experimental units each have their
own length of observation (which are not necessarily the
same). This would occur, for example, if the length of the
experiment was fixed but random entry into the experiment


4
was allowed. It is this latter type of censoring which this
work addresses.
Now we statistically formulate the problem of
I
interest. Let (X^,X2^) fr i=1>2,...,n denote a random
sample of bivariate pairs which are independent and
identically distributed (i.i.d.) and Ci i=l,2,...,n denote a
random sample of censoring times which are i.i.d., such that
C ^ denotes the value of the censoring variables associated
t I
with pair (X^,X2i). -*-n t^e case random right censoring,
the observed sample consists of (^ii>^2i^i) w^ere
I T
X j ^ = min(X,£,C^), X2j_ = min(X2i,Ci) and 6i is a random
variable which indicates what type of censoring occurred,
r
0i Description
1
2
3
4
Xli x;ici
Xli>Ci> x2i x;i>ci,x;i>ci
Now we state a set of assumptions which are referred to
later .
Assumptions:
I I
Al. (^ii>^2i) i=l2,...,n are i.i.d. as the
t
bivariate random variable
A 2
f t
(X^j,X2j) has an absolutely continuous bivariate
distribution function F(
X1
X2 U2
)


5
O
where F(u,v) = F(v,u) for every (u,v) in R The
parameters y ^ (p 2) and (a 2) ate location and
scale paratmeters, respectively. They are not
necessarily the mean and standard deviation of the
marginal distributions.
A3. Cj,C2*..,C are i.i.d. continuous random
variables, with continuous distribution function
G(c) .
A4. The censoring random variable is independent
f I
of i=l2 n and the value of is
the same for both members of a given pair.
A 5. P(xJi>C.,X2i>Ci) < 1.
A6. G (F ^ )) < 1 where F^ denotes the marginal
i i
T
cumulative distribution function (c.d.f.) of X^^
i=l,2.
Note that under A5, the probability is positive that the
sample will contain observations that are not doubly
censored.
With this notation, the null and alternative hypotheses
can now be formally stated. The null hypothesis is
Hq : y^=P2> 1=ct2 versus the alternatives:


6
1. The case where p^=y2=M with p known,
Ha: 1 a2
2. The case where pj=p2=U with p unknown,
Ha: i a2
3. Ha : p^ \i 2 and/or ^ a 2*
Chapter Two and Three will present test statistics for
alternatives 1) and 2). Chapter Four will present a test
for the more general alternative stated in 3). Monte Carlo
results and conclusions will be presented in Chapter Five.
First though, we describe related work in the literature.
Since this dissertation combines two areas of previous
development, that is, bivariate symmetry and censoring, the
first part of the review will deal with related works in
bivariate symmetry without a censoring random variable
considered. The second part of the review will mention
related works for censored matched pairs.
The first four articles to be considered, Sen (1967),
Bell and Haller (1969), Hollander (1971) and Kepner (1979),
all suggest tests directed towards specific alternatives to
the null hypothesis of bivariate symmetry. The work of
Kepner (1979) more directly influenced the development of
this thesis than the others, but they were direct influences
on the work of Kepner and thus will be mentioned.
Sen's article (1967) dealt with the construction of


7
conditionally distribution-free nonparametric tests for the
null hypothesis of bivariate symmetry versus alternatives
that the marginal distributions differed only in location,
or that the marginal distribution differed only in scale, or
that the marginal distributions differed in both location
and scale. The basic idea behind his tests is the
I I
following. Under Hq, the pairs (xii>X2i^ i=l2,...,n are a
random sample from an exchangeable continuous
distribution. He pools all the elements into one sample (of
size N=2n), ignoring the fact the original observations were
bivariate pairs and then ranks this combined sample. From
this, Sen obtains what he refers to as the rank matrix,
n / R11
R 1 2
R1 n
RN "
\ R 2 1
R 2 2 *
R2n
where R^^ is the rank of X^^ in the pooled sample j=l,2
i=l,2,...,n. Let S(R^) be the set of all rank matrices that
can be obtained from R^ by permuting within the same column
of R^ for one or more columns. Under HQ, each of the 2n
elements of S(R^) is equally likely and thus, if Tn is a
statistic with a probability distribution (given S(RN) and
H ) which depends only on the 2n equally likely permutations
of Rn, Tn is conditionally distri bution-free (conditional on
the given R^ and thus S(R^) observed). Sen's statistic Tn
can be defined as
T
n
n
n
l
i = l
,R
1 i


8
where ^ is a score function based on N=2n and i alone.
For the test of location differences only, Sen suggests
using the Wilcoxon scores (E^ ^ ) or the quantile F
scores (E^ ^ = F ^) where F is an appropriately chosen
absolutely continuous c.d.f.). The Ansari-Brad1ey scores
(En ^ ^ ~ |i ~ j|) or the Mood Scores
(Ew = ( 4 )^) are suggested for use when the
alternative is that the marginal distributions differ only
in their scale parameters. For the more general
alternative, that the marginal distributions differ in
location and scale, he recommends making a vector (of size
2) of his statistics where one component is one of the
statistics for differences in location and the other for
scale.
One basic weakness of Sen's proposals, as mentioned by
Kepner (1979), is that the procedure basically ignores the
correlation structure within the original observations
t
^Xli,X2i^ and, thus, suggests that a better test
could possibly be constructed by exploiting the natural
pairing of the observations.
The test proposed by Bell and Haller (1969) does
exploit this natural pairing of the observations. They
suggest both parametric and nonparametric tests for
bivariate symmetry. In the normal case, they form the
likelihood ratio test for the transformed observations
(^i i, Y2i) where Y1;L= X2i and Y2i = X1;L + X2i* The


9
resulting test they suggest when dealing with a bivariate
normal distribution is to reject Hq if |B ^ j > t(3^;n-2) or
j B 2| > t(g25n-l) where
(n-2)1/2 r(Y ,Y.) n^ Y
B. = and B =
1 2 l/ 2 c
(l-r'(Y1,Y2))/2 b
and r(Y^,Y2) is the sample correlation coefficient of the
2
Yj^'s and ^2i's ^ j and S are the sample mean and unbiased
sample variance, respectively of the Y^^'s and t(3;n)
represents the critical value for a t distribution with n
degrees of freedom which cuts off 3 area in the right
tail. The main problem with this test, as Kepner (1979)
also states, is that the overall level of the test, a, is
a = 23 1 + 2 3 2 43^
so relatively small values for 3^ and 32 would need to be
chosen .
The nonparametric tests they suggest are either
complicated, due to many estimation problems involved, or
have low power or are just unappealing due to the fact the
test is somewhat researcher dependent. (That is different
researchers working independently with the same data could
reach different conclusions.) Thus, they will not be
mentioned.
Hollander (1971) introduced a nonparametric test for
the null hypothesis of bivariate symmetry which is generally
appealing and consistent against a wide class of
alternatives. He suggested


10
D
n
/[ fFu-y)
- Fn(y,x)}2dFn(x,y)
where
is the bivariate empirical c.d.f. He notes that nDn is not
distribution-free nor asymptotically distribution-free when
Hq is true, and thus proposed a conditional test in which
the conditioning process is based on the 2n data points
(J i )
{((xHx2i)
V 0 or 1
for k = 1,2,...,n}
>
which are equally likely under HQ Here we let
(s,t)^) = (s,t) and (s,t)^^ = (t,s). This statistic
performs well even for extremely small sample sizes (n=5)
with one major drawback as mentioned by Hollander which is
the computer time which it takes to evaluate nDn. It
becomes very prohibitive for even moderate n. Koziol (1979)
developed the critical values for nDn for large sample
sizes, which work much better than the large sample critical
value approximations originally suggested by Hollander.
Kepner (1979) proposed tests based on the transformed
observations (Y^,Y2) of Bell and Haller for the null
hypothesis of bivariate symmetry versus the alternatives
that the marginal distributions differ in scale or that the
marginal distributions differ in location and/or scale. For
the alternative of differences in scale, he proposed a test


where
1 if t>0
'i'(t)
0 if t < 0 ,
which is Kendall's Tau applied to the transformed
observations. He noted that tt is neither distribution-free
nor asymptotically distribution-free in this setting and
thus recommended a permutation test which is conditionally
distribution-free based on tt for small samples. This
permutation test was based on conditioning on what he called
/
the collection matrix, Cn,
He noted that under HQ and conditional on Cn, there are 2n
equally likely transformed samples possible,
each being determined by a different collection of T* 's
j|f
where ¥^ = {1 or -1}. For larger samples, he obtains the
asymptotic distribution which can be used to approximate the
permutation test.
One nice property of the statistic tt which Kepner
notes, is that tt is insensitive to unequal marginal


1 2
locations and thus location differences do not influence the
performance of the test.
For the more general alternative of location and/or
scale differences, a small sample permutation test for
bivariate symmetry was proposed based on the quadratic form
whe r e
T
n
W + is the Wilcoxon signed rank test statistic calculated on
the Yj^'s and tt n is as previously defined. Again, the
conditioning of the test is on the collection matrix Cn. He
/
obtains the limiting distribution of the small sample
permutation test and proposes a large sample
distribution-free approximation which is computationally
efficient.
The second collection of articles which will be
mentioned deals with the topic of censored matched pairs.
Much work has been done recently in the area of censored
data, but the work of Woolson and Lachenbruch (1980) and
Popovich (1983) most directly influence the results in this
thesis and thus will be described here.
Woolson and Lachenbruch (1980) considered the problem
of testing for differences in location using censored


13
matched pair data. The situation they considered is
identical to the situation developed in this thesis if one
assumes equality of the scale parameters. They utilized the
concept of the generalized rank, vector introduced by
Kalbfleisch and Prentice (1973) to develop tests by
imitating the derivation of the locally most powerful (LMP)
rank test in the uncensored case. Although they imitate the
development of LMP rank tests for the uncensored case, it is
unclear whether these tests are LMP in the censored case.
Scores for the test are derived for (1) if the underlying
distribution the differences (i.e., X2^) is logistic
and (2) if the underlying distribution for the differences
is double exponential. In each case the statistic developed
reduces to usual statistic (Wilcoxon signed rank statistic
and sign test statistic for an underlying logistic density
or double exponential density, respectively) when no
censoring is present. Asymptotic results for the tests are
derived based on the number of censored and uncensored
observations tending to infinity simultaneously.
Popovich (1983) proposed a class of statistics for the
problem of testing for differences in location using
censored matched pair data. The class consists of linear
combinations of two statistics which are independent given
Nj and N2 where is the number of pairs in which both
members are uncensored and N2, the number of pairs in which
exactly one member is censored. The class of statistics can
be expressed in the general form of


14
T
n
(N1 ,N2 )
(l-L )V2
T, (N. )
In 1
Vo *
I/2 T (N.)
n 2 n 2
where T^n is the standardized Wilcoxon signed rank statistic
calculated on the uncensored pairs, and
T2n = N2_ /2
^N2R N2L^
where
N2R
i s
the number of pairs for
which Xis
censored
and X2^
is
not
, and N2L is the number
of pairs for which X2^ is censored and X^ is not (note
^2R+^2L= ^2^* The weight Ln is a function of and ^ only
P *
such that 0 distri bution-free statistic calculated only on the
uncensored pairs (and is a common statistic used for testing
£
for location in the uncensored case) while T2n is a
statistic based only on the type 2 and 3 pairs (as
previously defined in this introduction). The statistic
k
T2n is designed to detect whether type 2 pairs are occurring
more often (or less often) than should be under the null
hypothesis. Under Hq T2n is a standardized Binomial random
variable with parameters N2=n2 and p= V2 and thus
distribution-free. Popovich obtains asymptotic normality
for the statistic under the conditions (1) that
and N2 tend to infinity simultaneously and (2) under a more
general condition as n tends to infinity. In a Monte Carlo
study, he compares five statistics from this class to the
test statistic of Woolson and Lachenbruch (T^) (1980) based
on logistic scores. The results show that these statistics
perform as well as T WL (better in some cases) and that they


1 5
are computationally much easier to calculate. Furthermore,
exact tables can be generated for any member of the class
proposed by Popovich.
With the background established for the research in
this thesis, the attention will now be focused toward the
development of the test statistics to be investigated
here. Chapter Two will present a statistic for testing for
differences in scale which can be viewed as an extension of
Kepner's tt n for censored data. In Chapter Three, another
statistic will be presented for the same alternative but
more in the spirit of the work proposed by Popovich, that
is, the linear combination of two statistics which are
conditionally independent (conditioned on the number of type
1 and (type 2 + type 3) pairs observed). For the more
general alternative (i.e., differences in location and/or
scale), Chapter Four will present a statistic(s) which is a
vector of two statistics (one for scale and one for
location) following the work of Kepner. Lastly, Chapter
Five will present a Monte Carlo study of the statistics
developed in this dissertation.


CHAPTER TWO
A STATISTIC FOR TESTING FOR DIFFERENCES IN SCALE
2.1 Introduction
In this chapter a statistic will be presented for
testing the null hypothesis of bivariate symmetry in the
presence of random right censoring. Figure 1 represents a
possible contour of an absolutely continuous distribution of
this form. The alternative hypothesis for which this test
statistic is developed is : o^ t i.e., the marginal
distributions differ in their scale parameters. The
marginal distributions are assumed to have the same location
parameter. Figure 2 represents a possible contour of an
absolutely continuous distribution of this form.
The basic idea for this statistic was introduced in a
dissertation by Kepner (1979). He suggested the use of
Kendall's tau on an orthogonal transformation of the
original random variables to test for differences in scale
in the marginal distributions. The presence of a censoring
random variable was not included. To extend this idea to
include the presence of random right censoring, the concept
of concordance and discordance in the presence of censoring
which was used by Oakes (1982) was applied.
16


17
Figure 1. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales.


18
Figure 2. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales.


19
Section 2.2 will present the test statistic and the
notation necessary for its presentation. A small sample
test will be discussed in Section 2.3. Section 2.4 will
investigate the asymptotic properties of the test statistic,
with comments on the statistic following in Section 2.5.
2.2 The CD Statistic
In this section, the test statistic will be presented
which is designed to test whether the marginal distributions
differ in their scale parameters. First, since the work, is
so related, the test statistic which Kepner (1979) proposed
to test for unequal marginal scales will be presented. This
will give the reader an understanding of the motivation for
the test statistic.

Let for i=l,2,...,n denote independent
identically distributed (i.i.d.) bivariate random variables

which are distributed as (X,j,X2^)* Consider the following
t I
orthogonal transformation of the random variables (X^,X2^);
let
Yli = Xli + X2i and Y2i = Xli X2i for
Figure 3 illustrates what happens to the contour given in
Figure 1 (i.e., the contour of an absolutely continuous
distribution under HQ) when this transformation is


20
applied. Figure 4 shows what happens to the contour given
in Figure 2 (i.e., under H ) when this transformation is
applied. Note, as can be seen in Figure 3, under this
I
transformation and H Y, and Y01 are not correlated
o 11 11
t
although Xjj and X2 ^ possibly were. Similarly, as can be

seen in Figure 4, under this transformation and and
f
Y21 are correlated (negatively in this case). Thus, the
original problem of testing for unequal marginal scales has
been transformed into the problem of testing for correlation
I
between Y^ and Y2^ Kepner (1979) suggested the use of
T
Kendall's tau to test for correlation between Y^ and Y9y .
Kendall's tau was chosen, due to the fact it is a
U-statistic and, thus, the many established results for
U-statistics could be applied.
The test statistic which will be presented in this
section is very similar to the above mentioned statistic.
However, when censoring is present, the true observed value
If
of X ^ ^ 0^ X 21
(or
both) is not known,
a nd
thus
Y11 Y21
(or both) are
also
affected. To take
this
into
account a
modified Kendall's tau will be used which was presented by
Oakes (1982) to test for independence in the presence of
censoring. First though, some additional notation must be
int roduced.
I I
Recall, (Xjj,X2i) denotes bivariate random variables
t 1
which are distributed as (xii>^21^* Let 0^,02, Cn denote
the censoring random variables which are independent and
identically distributed (i.i.d.) with continuous


21
Figure 3.
Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales under the Transformation.


22
Figure 4. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales under the Transformation.


23
distribution function G(c) where denotes the value of
! t
the censoring variable associated with pair In
the case of random right censoring, the observed sample
f I
consists of X1 = min(Xj^,C^) and X2^ = mi n ( X£ ^ C ^ ) These
pairs can be classified into four pair types which are
Pair Type Description
f
1
Xli x2i *
2
Xii X2i> Ci
3
X¡i>Ci>
X2i 4
Xii>Ci>
X2i>Ci
Consider the following orthogonal transformation applied to
the observed sample:
/
Yli = Xli + X2i an(* Y2i = Xli X2i ^or i = 1 ^ . n.
Notice that, due to censoring in type 2,3, or 4 pairs, the
T
true values of and Y2^ (denoted Y^ and Y2i i.e. the
values had no censoring occurred) are not actually
observed. The following table, Table 2.1, summarizes the
t I
relationship of the true values of Y ^ and Y2^ to the
observed values


Table 2.1
24
Summarizing the Relationship Between the True
t
Values of and Y2^ to the Observed Values
Pair
Type
Description
Relationship
Between
Yli and Yli
Relationship
tBetween
y21 and Y2i
1 i X2i Yli
=
Yli
Y2 i
= Y2 i
li x2i>ci
YIi
>
Yli
Y2 i
< Y2 i
li>Ci
x2i 1
Yli
>
Yli
Y2i
> Y 2 i
1 i>Ci
X2i>Ci
Yli
>
Yli
uncertain
or Y2 ^ < Y2 i )
The modified Kendall's tau (denoted CD for
c^oncordant-discordant) can now be defined as
CD
a., b.. where for i < i ,
ij ij
r
r
1
if
Y
! Y, .
1
if
Y
v Y
l i
Ij
2 i
2j
<
-1
if
Yli '
Y. and
1 J
bu- <
-1
if
Y
2i
\ y
2j
0
if
uncertain of
0
if
unce r t ain of
V
the
relationship
V
the
relationship
(2.2.1)


25
(here Y11
I
can be read as
i
Y is definitely smaller
than
*- l_ *_ >
For example, if the iL pair is a type 1 and the j
pair is a type 2 and it was observed that Yli < Ylj. then
i t i i
aij = 1 since Y^ = Y^ and Y^j < Y^ (thus Y^ < Y^). If
Yli > Ylj had been observed, then a^^ = 0, since the
t f
relationship between Y^ and Y^j is uncertain. Similarly,
if Y2 < Y2j then f'ij = 0 since Y2 = Y2i and Y2j < Y2j
I
(thus, the relationship between Y2i and Y2j is uncertain).
On the other hand, if Y2i > Y2 j had been observed, then
b^j = -1 (by a similar argument).
Table 2.2 summarizes the necessary conditons for a^^
and b^j to take on the values of -1, 1 or 0. The product of
a^j and b^j results in a value of 1 if the it'1 and j*"*1 pairs
of the transformed data points are definitely concordant, a
value of -1 if the pairs are definitely discordant and 0 if
it is uncertain. If the iC^ pair is a type 4 (i.e., both
I
Xu and ^2^ were censored) then b^j will always be 0 since
the relationship between the i n and jcn pair is always
uncertain regardless of the pair's type. Thus, type 4
pairs always contribute 0's in the sum for CD. Notice, also
in the case of no censoring this modified Kendall's tau
reduces to the Kendall's tau applied to the transformed
data, the statistic investigated by Kepner (1979).


26
Table 2.2 Summarizing the Values of a. and b.. for i = 1: if and one of the following occurs,
ith pair type
Jth
pair type
1
1
1
2
1
3
1
4
a .
ij
= -1 :
if Y^ > Y^j and one
of
t he
following occurs,
t" Vi
i pair type
3th
pair type
1
1
2
1
3
1
4
1
a .
ij
= 0:
for all other cases
bij
= 1 :
if Y2^ < Y2j and one
of
the following occurs,
ith pair type
3th
pair type
1
1
2
1
1
3
2
3
bij
= -1 :
if Y2^ > Y2j and one
of
t he
following occurs,
ith pair type
3 th
pair type
1
1
1
2
3
1
3
2
bU
= 0:
for all other cases


27
Next, we establish some properties of the CD statistic.
Lemma 2.2.1: Under HQ ,
E ( C D ) = 0
and
Var(CD) = a +
n ( n 1)
n ( n 1 )
where
a nd
a = 4P(a .= 1,b. = 1 )
ij iJ
2p bi j =
1 a . =
ij
f N
*H
II
*-)
*H
x¡
+ ZPia.j.
= 1 a . ,
ij
= -1,b . =
ij
1)
+ 4P(a1.=
= -1,b . =
ij
1)
- 2P(a1j =
l.bpj-
= 1,b. =
ij
1)
- 2PUlj-
-l.bij
= -1 a .
ij
-i,bir
= 1)
- 4P(a1;j-
'bij"
1 "

1)
(2.2.2)
Proof:
Throughout this proof, Theorem 1.3.7 in Randles and
Wolfe (1979) will be used extensively and thus its use will
not be explicitly indicated.
Under H
>
(Xli,X2i,X1j ,X2j ,Ci ,Cj ) = ^X2i,X1iX2j X1j Ci Cj^


28
and therefore it follows that
(XliX2iXlj X2j 0i>6j} (X2i,Xli,X2j ,X1 ,f(6 ) ,f(6j ) )
(2.2.3)
where
Xu = min(xJi,C1),

X2i = rain(X2i ,C),
^ indicates what type of pair (X^.X^)
and
f(6i) indicates what type of pair (X2i,Xli^ is
Thus, f(*) is the function defined below.
5i
f (6)
1
3
2
4
Let Yli Xli + X2i a nd Y2i Xli X2i; thus from (2.2.3)
(YliY2iYljY2j6i>6;j> = (Yli,-Y2.,Ylj,-Y2j,f(6i),f(6j))
Applying the definition of a^ and b ^ j in (2.2.1) (or using
Table 2.2) to the above, it follows that
and thus
p

29
and
p (ai j -1> bij Pbij -1J
(2.2.4)
Now,
E ( C D ) = I E ( a . b . )
(?) i where
ECa^bjj) (DF^jbjj 1) + (-I)P(aijblj -1)
p - P(atj l.bjj -1) P(ai:j -l,btj 1) .
Applying (2.2.4) to the above, it follows that E(a^^.b^j) = 0
and thus E(CD) = 0.
Note, that under H Q ,
(Xli,X2i*Xlj ,X2j <5i,<5j) ^Xlj X2j xiix2i5j > and thus
(YlfI2i.'ilj-y2J-Si.5j>
(2.2.5)
Applying the definition of and b^j as before, it follows
(au>bij) (-aij-'bij)
and also


30
and
P(a13 l,b13 1) P(a^j -l.bij -1)
P p Now,
Var(CD) = [] Var( £ a b..)
' ^ 1 i !J 1 J
(J) ^<3
, 2
= [ ] J y Cov ( a . b . a , b )
r?i i The three possible cases to consider for the covariance are
1 ) ii j i 2) i-i' j=j i< j i < j '
and
3) where exactly two of the four subscripts
i Case 1) i/i j tj' :
In this case, Cov(a^jb^j, a^ij,b^iji) = 0 since the
bivariate pairs are i.i.d.
Case 2) i=i', j =j':
In this case ,
Cov(aijbij at j bjj ) EU.jjby) ]
P(aij >bij > + P(aij 1 bij -*>
+ P> + P = 4 p(aij = 1 t> i j = 1) = 4a (by part a).


31
Case 3) Exactly two of the four subscripts i the same.
Now,
Cov(aij bj
aikbik)
- E(atJb
i j aik
the following
events:
Al,i:
a i j "
1 aik =
1}
Al,-1:
{aij =
aik =
-1}
A-1 1:
{aij =
_1 aik
= 1}
A-1,-l
: {aJ =
-1 aik
= -1}
and similarly define the events B_^ ^ and
B_j Using this notation, E ( a ^ ^ b ^ j a ^ ) can be written
as
E(aijbijaikbik) =
l I l I (-l)k+I+"+"p(A
k = 0 A-0 m = 0 n =0 (-1) ,(-1)*
,B )
(-l)m,(-1)n
(2.2.6)
Table 2.3 describes the events A and B
<-Dk,(-i)t <-i)m,<-nn
in more detail and the restrictions placed on the 6's.
Now, to simplify the probabilities in (2.2.6). Note,
under Hq
d
X2j ,XlkX2k,,Si5j ^ k ^
X2k>Xlk>f(6i)>f


32
Applying the transformations
Yli = Xli
X2i and Y2i Xli X2i
it f ollows that
6i.6j'Sk>
Now, applying the definitions of a^j and in (2.2.1)
using Table 2.2), notice that if b^j = 1 (i.e.,
6 e(1,2) and 6j£(l,3), then ~Y. > -Y. f(6 ) e( 1 3)
f(6j)e(l,2) which would yield b^j = -1.
Using similar arguments, it follows
and
(a..,a.,,b..,b.,) = (a..,a..,-b..,-b.,)
ij ik ij lk ij ik ij lk
and thus
P(A,
1 9
1
,B 1,1 ) =
p(Ai,i
-1
P(A,
* 9
1
,B-1,1)
- P (A.
1 9
1 '
B1
P(A_1
9 ~~
1 ,B 1 1 )
= P(A_
1-1
9
P(A_j
9 ~~
1 *B-1,1
) = P (A
-1 ,
-1
B 1 1
P(A_X
, 1
,B 1 1 )
= P(A_1
, 1
P(A_j
, 1
B-l,1}
= P (A_
1 1
)
P ( A ,
1 9
-1
B 1 1 ^
= P(A,
1 9
-1
> B
-i.-i
P ( A,
L 9
-1
B-l,1)
= P(Aj
9
Bi,-i>
(or


Table 2.3 Describing the Events A and B and
Event Description
A1,1B1,1
CYll (Y21 A-1, 1B1,1
(Y11>Y1jYli
A1,-1B1,1
Yik)
A1, 1B-1,1

Y2j
A1,1B1,-1

A-1,-1B1,1
(Yjl>Ylj,Yli>Yjk)
A-1,1B-1,1
Y1j *Yli
(Y2i>Y2j
A-1,1B1,-1
(Y1i>Y1J,Yii A1,-1B-1,1
Ylk>
Y2j
A1,-lBl,-l
Ylk>
A1,1B-1,-l
^ Yli Y2j
A-1,-1B-1,1
(YU>YljYli>Ylk>
Y2j
A-1,-1B1,-l
(Yl1>Yjj,Y,1>Yik)
(Y2i A-1, 1B-1,-1
Ylj-Yli
(Y2i>l2j
A1,-1B-1 ,-l
Ylk)
Y2j
>
1
1
cc
1
1
H-
Ylj-YU>Ylk>
Y2j
the Restrictions on the
6 s
Restrictions
on the
6 s
6 j
Y2i
1
1,3
1,3
Y2i 1
1
1,3
Y2i
1
1,3
1
Y2i
1
1,2
1,3
Y2i>Y2k>
1
1,3
1,2
Y21
1,2
1
1
Y2i 1
1
1,3
Y2i> Y2 k >
1
1
1,2
Y2i 1
1,2
1
Y2i>Y2k>
1
1,3
1
Y2i>Y2k)
1
1,2
1,2
Y2i 1
1
1
Y21>Y2k>
1
1
1
Y2i>Y2k>
1
1
1,2
Y2i>Y2k>
1
1,2
1
Y2i>Y2k>
1,3
1
1


34
Similarly, under HQ
(Xli,X2i,Xlj,X2j,Xlk,X2k,<5i5jlSk^
= (xiiX2i>xlk>X2kXljX2j6i>6k6j)
and applying the definition of a^ and b^j in (2.2.1) it
follows that
(aij >aikbij >bik) (aikaij )bikbij}
This yields that
P B-1, i^ = p(Ai,-i
B 1 -1
*-d
>
1
*
Bi ,-P = p(Ai ,-i
B 1 1
p ,B i i ) = P(A_1 ^l
B1 1 )
p B -1 ,-P = P(A-1 ,
1 B-1,
Thus, E(ajb^jaikbik^ can re^uced to a sum of six terms
instead of the original sixteen; i.e.
E(ab..ab ) =
ij ij ik ik
1111
l l l l (-D
k = 0 £ =0 m = 0 n =0
k+£+m+n
p (A. . ,B )
(-i)k,(-i)* (-n,(-Dn
2P(A1,1 + 2P(A-1,-1 Vl> + 4P(A1,-1
' 2P 4P(A1,-1B1,1> 4
Note, the subscripts are arbitrary; thus
E(aijbijaikbik) = E(aijbijakjbkj) = E(aijbijajkbjk)


35
and therefore combining the results from case 1, 2 and 3, it
follows that
!-l ) Y}
Var(CD)

As seen in Lemma 2.2.1, the variance of CD depends on
the underlying distribution of and possibly C.
Therefore, CD is not distribution free under HQ. Section
2.3 will discuss a permutation test based on CD that is
conditionally distribution free. This test is recommended
for small samples. For larger samples, Section 2.4 presents
the asymptotic normal distribution of CD using a consistent
estimator of the variance. This result can be used to
/
construct a distribution free large sample test based on CD.
2.3 Permutation Test
In the situation where the sample size is small, a
permutation test based on CD is recommended. What is
considered a small sample size will be discussed in Chapter
Five when the Monte Carlo results are presented. Now, we
will develop the motivation for the permutation test.
Recall, under HQ
f
(X
2i,Xli>Ci>


36
and thus
x2i
, <5 i) = (X2i,Xli,f(6i))
(2.3
where
Xli =

min(Xji,), X2i
= min(X2i,Ci), 6^ is the
pair
type
( i e .
6 ^ = 1,2,3 or 4)
and f(<5^) is a function
such
that
1
2
3
4
f (5)
1
3
2
4
0
Let k = | j be an operator such that
r
(XliX2i6i)k = \
(X1iX2i6i)
if k = 1
(X2iXlif(6i)) if k = 0
and K = {k: k is a 1 x n vector of 0's and l's} (of which
there are 2n different elements). Thus, applying this
operator to (2.3.1), we see under HQ, P{(X^,X2^,6^) =
(Xu,X2i,6i)0} = P((Xli,X2i,6i) = (X1.,X2i,6.)1}. Applying
this idea to the entire sample (in which the observations
are i.i.d), under H it follows that
{(Xll*X21^i^ ^(^X2X22^2^ ^ ^XlnX2n,<^n^t1^
t I
^91 *6}) 1 ,(X.2,X22,62) 2,...,(X.,X2n,6n) n}
(2.3.2)



37
where k and k' are arbitrary elements of K. Therefore,
unde r HQ, given
{ (x
11 x21^1^ >(xi2,x22^2^* * ^ x1nx2 n 6n)>,
the 2n possible vectors
{ (xiix21^l) l(xi2x2 2^2^ ^^xlnx2n^n^ n}
are equally likely values for
{ ( X j i >X£i <5 ^ ) ,(Xi2>X22><52^ * ^ln^2n^n^ *
The idea of the permutation test is to compare the
observed value of CD, for the sample witnessed to the
conditional distribution of CD derived from the 2n equally
likely possible values of CD (not necessarily unique)
calculated from
{ (xii ,x21 1>(x12x22^2^ 2,...,(xin>X2n(5n^ n ^
Note, since the sample observed is censored, the 2n
vectors {xii>x21^l^ l,(x^2x22^2^ ^''^xlnx2n^n^ n }
are not necessarily unique. If a pair is a type 4 (i.e.,
both and X2j were censored), then (xij >x2j >^j ) ^ =
(xjj,x2j 6j In fact, there are only 2^n-n4^ unique
vectors (n^ = number of type 4 pairs), since P(Xj^ = ^2^) =
0 if ^xii^2i^ is not a cYPe ^ pair under assumption A2. As
a result, the permutation test, in effect, discards the type
4 pairs (since a j b j = 0 if the i c or j C ^ pair is a type
4) and treats the sample as if it were of size n-n^ with no
type 4 pairs occurring.
With regards to the transformed variables (^ii*^2i^
i = 1,2 the permutation test can be viewed in the
following way. Consider the transformations = X^ + X2^


38
and Y2^ = ^2i* Applying these to (2.3.1) and (2.3.2),
we see that under Hq
(Yli,Y2i,6i) = (Yli,-Y2i,f(6i))
and similarly,
{ ( Y
11 L21
Si)
( Y
1 2 12 2
s2)
( Y
In 2 n
5n>
n-,

!
k
whe re
k.

if k = 1
= ^
(YliY2if(6i}) if
k = 0
and k and k' are arbitrary elements of K. That is, under
H0, given { (y:2 ,y21,Sx),(y12,y22 s2),..., (yln,y2n,sn)}, the
2n possible vectors
kl k2 kn
^yl1 y2161) >(yi2y22^2^ *^ylny2n^n^ } are
equally likely values for
{ (Yii Y2 i ^ i ) ,(Y12>y22,S2) . >(Y^n,Y2n,6ri)}
To perform the permutation test, the measurements
(x1i*x2i^i) ^ = l>2,...,n are observed and the
corresponding value of CD is calculated. Under Hq, there
are 2n equally likely transformed vectors for
{(YllY216l)(Y12Y2262)*(YlnY2n

39
statistic is computed for each of these possible vectors and
from this the relative frequency of each possible CD value
is determined. The null hypothesis is rejected if the
original observed CD value is too large or too small when
compared to the appropriate critical value of this
conditional distribution.
2.4 Asymptotic Results
In Section 2.3, a permutation test was presented to
test Hq, when the sample size was small. In larger sample
sizes, the permutation test becomes impractical and time
consuming. In these situations, the asymptotic results
which will be presented in this section could be employed.
Theorem 2.4.1: Under Hq,
* N(0,1) as n * ?
[ Var(CD) ]X/2
where
Var(CD)
2 4(n-2)
n(n-l) a n(n-1) Y
Proof;
Note that CD is a U-statistic with symmetric kernel
h(X^,Xj) = Thus, by applying Theorem 3.3.13 of


40
Randles and Wolfe (1979), it follows that
CD
+ N(0 ,
as n > oo
whe re
?1 = E[h(X.,Xj)h(Xi,Xk)]
Note that
= E[aijbij aikbik]
il
n
2 4(n-2)
n(n-l) a n(n-l) ^
Y
as n + <*>,
therefore after applying Slutsky's Theorem (Theorem 3.2.8,
Randles and Wolfe, 1979)
CD
[Var(CD)J^2
+ N(0,1) as n + .

Corollary 2.4.2: If Var(CD) is any consistent estimator of
Var(CD), then
+ N(0,1) as n -* .
[ Var (CD) ]^2
Proof;
This follows directly from Theorem 2.4.1 and Slutsky's
Theorem. ^
Next, we consider the problem of finding a consistent
estimator for Var(CD). There are many consistent estimators


41
for a variance, but three which worked well in the Monte
Carlo study are described in the following lemma.
Lemma 2.4.3: Under Hq the following are consistent
estimators of Var(CD):
1) Va r x(CD)
4
n
lllA
1 ijk
y
where
AijkBijk = (aijbijaikbik + aikbikajkbjk + aijbijajkbjk)>
2) Var2(CD)
- 4 (
l l l + l I 1 > >.
n n(n-l)2 l and
3)Var3(CD)
<[tj- l l 21
n(n-l) ( 2 ) 1 4(n-2) rn .N.
+ {- Var2(CD)}
n(n-1) 4
Proof:
First, it will be shown that nVar^(CD) 4y.
nVa r ^(CD)
lllA
1 < i < j i jkBij k
}
Now,


42
, A B .
- Hr I l l
1 which shows that nVar,(CD) = 4U where U is a U-statistic
J- n n
of degree 3 with symmetric kernel h* = A^^^3^^^/3 Thus, it
follows that nVar^(CD) 4y since y by Hoeffding's
Theorem (Hoeffding, 1961).
Next, it will be shown that
n(Var2(CD) Var^CCD)) 0 as n .
First though, notice that Var2(CD) is equivalent to
2)Var1(CD) + -{ [ l l (a b )2] (CD)2}
(n-1) n n(n-l) l Thus ,
(n-2 )
n(Var2(CD) Var^CD)) = n{ f 1} Var^CD)
(n- 1 )
+ 4{[
l l (a b )2] (CD)2}
n ( n-1 ) 2 1 < i < j < n 1 ^
(n-2 )
(n-1)
- U B* + {[ L l J (a b )2] (CD)2}
(n-1)(2) 1 0 as n > oo.
Therefore, Var2(CD) is a consistent estimator for Var(CD).
Lastly, it will be shown that
n(Var^(CD) Var2(CD) ) * 0 as n + .


43
Now,
n(Var3(CD) Var2(CD))
-L- [tst I l (aiibii)2] (n-1) (") 1 + {(n 2) 1} nVar2(CD)
(n-1)
-> 0 as n -*

Next, we provide a brief explanation of each of these
estimators. As was shown in the proof of Lemma 2.4.3,
^ 4 *
Var,(CD) = U where U is a U-statistic which estimates y.
i n n n
Thus, Var^(CD) is estimating the asymptotic variance of
CD. Var2(CD) is also estimating the asymptotic variance of
CD, but in a slightly different manner. Recall, from basic
U-statistic theory that y is the variance of a conditional
expectation (Randles and Wolfe, 1979, p. 79) (i.e.,
y = Var[(ajbj)*] where (ajbj)* = E[a12b^2¡ (Y11,Y21)]) .
Thus, in Var2(CD), for each (^j^,Y2^), the conditional
expectation is estimated using all the other (Y^,Y2j)'s,
j*i and then the variance of all these quantities is
calculated. That is,
where
Var-(CD) = Y {(a.b.)* CD}2
4 n t l l
n-1 j*i J
In contrast to Var^(CD) and Var2(CD), Var3(CD) is
estimating the exact variance of CD (2.2.2) derived in


44
Section 2.2. It is using an estimator of y from Va^CCD)
and estimating a with a difference of two U-statistics which
is estimating
Again, although under HQ ECa^b^j) = 0, the sample
estimate for Eia^b^j) (i*e., CD) was left in to possibly
increase the power of the test under the alternative.
Each of these variance estimators will be considered in
the Monte Carlo study in Chapter Five. Although the
calculations look overwhelming if performed by hand, they
are all easily programmed on the computer. (See the CDSTAT
subroutine in the Monte Carlo program listed in Appendix 2.)
2.5 Comment s
This chapter has presented a statistic to test the null
hypothesis of bivariate symmetry versus the alternative that
the marginal distributions differ in their scale
parameters. For small samples, a permutation test is
recommended. A basic disadvantage of this is that it
generally requires the use of a computer for moderate sizes
(otherwise it is very time consuming to derive the null
distribution). For larger sample sizes, it is recommended
C D
that be used as an approximation for
[Var(CD)f^2
CD
Thus, for an a level test using the
[Var(CD)]1/2


45
asymptotic
rej ected if
distribution,
i CD
^ Vo
[Var(CD)] 2
the null hypothesis would be
> Z
a/2
where Z is the value in
a / 2
a standard normal distribution such that the area to the
right of the value is a/2.
Chapter Five will present a Monte Carlo study which
uses the asymptotic normal distribution of CD (with a
consistent variance estimator) to investigate how well the
test performs under the null and alternative hypotheses.
First though, some comments on this chapter.
Comment 1
One possible advantage of the CD statistic is the fact
it utilizes information between censored and uncensored
pairs whenever possible. In the permutation test, type 4
pairs have no effect on the outcome of the test. That is,
they can be ignored, treating the sample as if it were of
size ni+n2+n3* This is understandable since Xli = X2i = Ci
and thus they supply no information about the scale of
relative to
In the asymptotic test, if one estimated the variance
in (2.2.2) by estimating a and y with their sample
quantities (for example, a =
n(n-l) 1 < i < j < n
l l (aijbij) ic


46
is easily shown that the type 4 pairs have no effect on the
value of the test statistic. That is, the value of the test
statistic remains the same whether the type 4 pairs are
discarded or not. If a different estimate for the variance
is used, there is a slight change in the test statistic's
value if type 4 pairs are discarded, due to the different
variance estimator. Asymptotically, this difference goes to
zero, due to the fact the variance estimates are all
estimating the same quantity. Thus, in some sense, the
asymptotic test behaves similarly to the permutation test
with regards to type 4 pairs.
If a and y are known, they are a function of whether
type 4 pairs are included or not. That is, if type 4 pairs
were not included in calculating the test statistic (thus
n=n^+n2+n3), the value for a and y would be larger than the
value had type 4 pairs been included (since type 4 pairs
only contribute 0's and never l's or -l's). The effect of
type 4 pairs on a and y is such that the test statistic's
value would be the same (or at least asymptotically the
same) whether type 4 pairs were discarded or not.
Comment 2
A disadvantage of the test is that for small samples CD
is not distribution free. Thus, the permutation test,
conditioning on the observed sample pairings, must be


47
performed to achieve a legitimate distribution free crlevel
test .
Comment 3
It is unclear how the CD statistic would be affected if
the marginal distributions of and X21 have different
locations. It is possible that the assumptions made on the
censoring distribution might not be valid (in particular
assumption A4, which assumed the same censoring cutoff for
X^ and ^2i^ or even if this is true, that CD does not
perform well in these instances. Chapter 5 will investigate
this problem in further detail


CHAPTER THREE
A CLASS OF TESTS FOR TESTING FOR DIFFERENCES IN SCALE
3.1 Introduction
In the previous chapter, a test statistic was presented
to test the null hypothesis of bivariate symmetry against
the alternative that the marginal distributions differ only
in their scale parameters. A shortcoming of the statistic
was the fact the variance of CD depended on the underlying
distribution and, thus, for a small samples a permutation
test had to be done or for large samples the variance had to
/
be estimated. In this chapter, two test statistics will be
presented which are nonparametrica1ly distribution-free
(conditional on N^ = n^ and Nc = n2+n^) for all sample sizes
to test the null hypothesis of bivariate symmetry. The
alternative hypotheses are structured by assuming the
samples come from a bivariate distribution with c.d.f.
*1 U x2 p
F( ) where F(u,v) = F(v,u) for every (u,v)
a 1 2
2
in R Tests are developed for both of the following
alternatives to the null hypothesis of bivariate symmetry:
48


49
Case 1 = ^2 known,
Ho: ai = a2 an<^ Ka: o< a 2
That is, the marginal distributions have the same known
f
location parameter but, under Hfl, X2^ has a larger scale
I
parameter than X^ A possible contour of an absolutely
continuous distribution of this form was given in Figure 2.
Case 2 p j = pi 2 unknown,
a 1 = a 2 and Ha
a L < a 2
Here, the marginal distributions have the same unknown

location parameter but, under Ha, X2 ^ has a larger scale
parameter than X^
(Note, for both cases, the alternative has been stated in
the form for a one sided test. The procedure which will be
presented can easily be adapted for the other one-sided or a
two sided alternative. The latter is discussed at the end
of this chapter )
In Sections 3.2 and 3.3, tests statistics for Case 1
and Case 2, respectively, will be presented which are
nonparametrically distribution-free conditional on N ^ = n ^
and Nc = n9+n^. In both cases, the test statistics can be
viewed as a linear combination of two independent test
statistics T and T where T is a statistic based only
nl nc nl
on the n^ uncensored observations, while Tn will be a
statistic based on the n£ = n2 + n3 type 2 and 3 censored
observations. The conditioning of the random variables


50
and Nc on n^ and n2+n^ (respectively) is used throughout
Section 3.2 and 3.3 and, thus, this condition will not
always be stated but will be assumed with the use of n^, n2
and n7. Thus, the test statistics will be written as T
3 ni n(
and TMn n (for Section 3.2 and 3.3, respectively) which
1 c
imply conditioning on = n^ and Nc = nc = t^ + n^. Section
3.5 will consider the asymptotic distribution of each test
statistic .
3.2 p ^ = p 2 > Known
This section will begin by introducing the notation
necessary for the statistic Tn n designed for the
1 c
alternative in case 1. Recall, the sample consists of
(Xli,X2i) i = l>2,...,n where = min(X^,C^) and
I
X2i = n(^2 iCi). These pairs were classified into four
types .
They
were the
following

Pair
Type
Description
Number of Pairs
in the Sample
1
X1i X2i nl
2
Xli X2i>Ci
n2
3
Xii>Ci
X2i n3
4
xU>ci.
X2i>Ci
n4
n^ + 02 +
where n
n3 + n4


51
For convenience and without loss of generality, let the
type 1 pairs occupy positions 1 to n^ in the sample (i.e.,
{(XjJ,X21),(Xj2,X22(Xln ,X2ni)} ) in random order.
Similarly, the type 2 and type 3 pairs will be assumed to
occupy positions n^+1,n^+2,...,n^+nc in random order.
Lastly, the type 4 pairs occupy positions
n^+nc+l,n^+nc+2,...,n. What is meant by random order, is
that the exchangeability property still holds within the n^
type 1 pairs, within the n2 + n2 type 2 or 3 pairs and within
the n^ type 4 pairs. This could be accomplished, if the
pairs were placed into their respective grouping (type 1, 2
or 3, or 4) arbitrarily, with no regard to their original
position in the sample. Much easier, from a researchers
point of view, would be to place the pairs into their
respective groupings in the same order they occurred in the
sample (i.e., the first uncensored pair is placed into the
first position among the n^ uncensored pairs, the second
uncensored pair into the second position, etc.) This
procedure would not affect the desired exchangeability
property, as deduced from the following argument. In using
the second method, the reseacher is actually fixing the
position of the type 1 pairs, type 4 pairs and type 2 or 3
pairs. Thus, instead of n! equally likely arrangements of
the original sample, there are n^!n^!(n9+n2)! equally likely
arrangements when the positions and numbers of the pair
types are fixed. Therefore, it follows, that each of the
n^! arrangements of the n^ uncensored pairs is equally


52
likely and that the exchangeability property still holds
within the type 1 uncensored pairs. Similar argueraents for
the ^2 + 1*2) type 2 or 3 pairs and the n^ type 4 pairs
hold .
The following notation will be used in the statistic
Tn^, a statistic which is based on the n^ type 1 pairs.
Define a variable to be
Zi I X 2 i y I I X1 i 11 I for i_12...n1
where p is the known and common location parameter. Let
be the absolute rank of for i = 1 2 . n ^ that is, the
rank of j Z ^| among {|z^|,|Z2|,...,|zn j} and let be
defined as
H'i = 'R C Z ) =
1 if Z > 0
0 if zi < 0
Note, the variable Z ^ is defined only for the uncensored
pairs
The statistic T is then
nl
ll
: = y i.
n. L 1
R
i = l
the Wilcoxon signed rank statistic computed on the Z^'s.
Notation will now be introduced for the statistic Tn ,
c
a statistic based only on the type 2 and type 3 censored
pairs. (The pairs in which only one member has been
censored.) Define Qj to be the rank of Cj among
^ Cn1 + 1Cn1+2 ,Cni + nc* and


53
YJ 1
h
1 if the j pair is a type 2 pair
0 if the j11*1 pair is a type 3 pair
for j = n^+1,n^+2,...,n,+n^. The statistic Tn is defined
1 c
a s
n, + n
1 c
C = l y. Q.
nc j=n1+l J J
= y ranks of the C's for the type 2 pairs
A brief explanation of the logic behind the test
statistic will be presented. For the test statistic Tn if
X2 has a larger scale parameter than Xj (i.e., under ) ,
then 1^2^ pJ |x^ p| should be positive and large.
Thus, the test statistic Tn would be large. In contrast,
if X2 and X: have the same scale parameter (i.e., under HQ ) ,
then I X 2 ^ ~ pI ~ |x^ p| would be positive approximately
as many times as negative with no pattern present in the
magnitudes of 1x2^ p| |x^ pj. Thus the test
statistic would be comparatively less.
For the test statistic T if H is true, there should
nc a
be a preponderance of type 2 censored pairs (relative to the
number of type 3 censored pairs) and these pairs should have
the more extreme censoring values. Figure 5 illustrates
this idea. Thus, the test statistic Tn would be large. In
contrast, if H is true, the number of type 2 pairs should


54
Figure 5 .
Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales with Censoring Present.


55
not dominate n and the test statistic T should not be
c nc
unusually large.
Now we establish certain distributional properties for
T_ and T_ .
n. n
Lemma 3.2.1:
distribution
Conditional on n^ T has
as the Wilcoxon signed rank
the same null
statistic.
Proof:
First it will be shown that conditioning on the
1 pairs does not affect the exchangeability property
(XliX2iCi) (X2iXjiCi)) still holds. Let
(XiX¡i.Ci) and W* = (X2i,x|i,Ci) and
G (t) =P(xJt< t x X2i^ t2Ci< t3) = E[I(W < £)] where
r 1
i(w < t)
i
0
if X..< t
11
otherwise
X. <
2i
t2Ci<
nl type
( i e ,
Now, under HQ for the entire sample, we have
(Xii>X21>Ci> Cl>
and applying an apropriate function (and Theorem 1.3.7 of
Randles and Wolfe, 1979) thus
I(W. < t)I(5i= 1) = I(W* < t)I(f(6.) = 1) .
Taking expectations, it follows that
E[I(W.< t)I(6.= 1)] = El I(W < t)I(f(5.) = 1)] .
Now, recalling that 5^= 1 iff f(5^) = 1; thus
E[ I ( 5 i = 1)] = E[ I ( f ( <5 i ) = 1)] ,
and it follows that


56
E [ I( W < t)I(6jL= 1)] E [ I (W* < t)I(f(6i) = 1)]
' ~" 1
E[I(5.= 1)] E[I(f(5) = 1)J
This shows that the c.d.f. of given it is a type 1 pair
is equal to the c.d.f. of Wi given it is a type 1 pair and
thus the exchangeability property holds within the type 1
pairs.
Now, by defining a function
fj(a,b,c) = |min(b,c) y| |min(a,c) y| and applying
Theorem 1.3.7 (Randles and Wolfe, 1979, page 16) it follows
Z | X 21 ~ y |
- 1X11 l
= |min(X21>C) yj
j
- min ( X C )
d | ,
i
- |min(X2^ ,C)
= |min(X1j C) y 1
= 1 X1 l n| -
1X21 ~ =
and thus by Theorem 1.3.2 (Randles and Wolfe, 1979, page
14), the random variable Z has a distribution that is
symmetric about 0. The proof of Lemma 3.2.1 follows
directly from Theorem 2.4.6 (Randles and Wolfe, 1979, page
50) D
Lemma 3.2.2: Under HQ, the following results hold.
a) Conditional on the fact the pair is type 2 or 3, the
random variables Yj and Cj are independent.
b) Conditonal on nc, Tn has the same null distribution
as the Wilcoxon signed rank statistic.


57
Proof :
First, it will be shown that conditioning on the n c
type 2 and 3 pairs does not affect the exchangeability
property. Define W^, GW^ £ ^ and I(W^ < t) as in Lemma
3.3.1. Now under HQ for the entire sample, we have
(^1 i,X2i ^i) = (^2iXli>^i^
and applying an appropriate function (and Theorem 1.3.7 of
Randles and Wolfe, 1979)
d *
I(Wt < t)I(5ie(2,3)) = I(W. < t)I(f(5i)e(2,3)).
Taking expectations, it follows that
E{l(Wi Recalling that, 6^e(2,3) iff f(6^)e(2,3), and thus
E[I(5ie(2,3))J = E[I(f(6i)e(2,3))].
It follows that
E[I(W.< t)I(ie(2,3))] E[I(W*< t)I(f(6 ) e(2,3 ) ) ]
E[I(5ie(2,3))] E[I ( f ( 5 ) e ( 2 3 ) ) ]
Therefore, conditional on the pair being a type 2 or 3, the
exchangeability property still holds.
Thus, it follows that
P(Yj = l.Cj < c) = P(xJj < > Cj.Cj < c)
= Cj^lj ^ ^ c) = P ( Y j = 0>Cj c)
Noting that,
P(Yj = l.Cj < c) + P(Yj = O.Cj < c) = P(Cj < c)


58
and thus
2P ( yj = l.Cj < c) = P(Cj < c)
or that
P ( Y j = l.Cj < c) = V2 P (Cj < c) = P ( Y j = 1 ) P (Cj < c)
and thus we see that Yj and Cj are independent.
To prove part b), let y = (Yn +1 >Yn +2 Yn +n
1 1 1 c
and
Q = (,0n +2 ,...,Qn +n ). By Theorem 2.3.3 (Randles
11 1 c
and Wolfe, 1979, page 37), Q is uniformly distributed over
Rn where
c
Rn = {q : q is a permutation of the integers 1,2, ....n^ }.
Now, let q be any arbitrary element of Rn and let g be any
c
arbitrary nc vector of 0's and l's. Thus,
P (Y = g >0 = 5) = P(X = g ) P (Q = <{ ) (by part a)
and
P(Y = g ) P (Q = <{ ) = x
- C n !
2 c
which proves part b). £3
By Lemmas 3.2.1 and 3.2.2, TR and Tn are
1 11 c
nonparametrica1ly distribution-free conditional on n^ and
nc, respectively.


59
Lemma 3 2 .3 : Under HQ, the following results hold,
a) Conditional on n^ E(Tn^) = n^(n^+l)/4 and
Var(Tn^) = nj(nj+1)(2nj+l)/24
b) Conditional on n E(T ) = n (n + l)/4 and
c n c c c
Var(Tn ) = nc(nc+l)(2nc+l)/24.
c)
Conditional on n, and n .
1 c
independent.
T and T are
nl nc
Proof:
The proof of parts a) and b) follow directly from
Lemmas 3.2.1 and 3.2.2 and the fact that the Wilcoxon signed
rank statistic based on a sample of size n has a mean of
n(n+l)/4 and variance of n(n+1)(2n+l)/24.
The proof of part c) is also trivial following from the
fact Tn and Tn are based on sets of mutually independent
observations. ^
With these preliminary results out of the way, the test
statistic T can now be defined by
1 c
T
nlnc
L1 T
1 nl
+ L2Tn
nl
n l + nc
y 'v. r .
i=i 1 1
+ L2
+
ii
II
)
where Lj and L2 are finite constants.


60
Theorem 3.2.4: Under HQ ,
> E L1E + E2E(Tn >
i c i c
= (LjiijCnj + l) + L2nc(nc + 1 ) )/4
b) Var(Tn n ) = (L^(nj + 1)(2ni+l)
1 c
+ L|nc(nc+l)(2nc+l))/24
c) Tn n is symmetrically distributed about E(Tn n )
and
d) for fixed constants Lj and L2 Tn n is
nonpararaetrically distribution-free .
Proof :
The proof of parts a) and b) follow directly from
Lemmas 3.2.2 and 3.2.3. To prove part c), it is known that
the Wilcoxon signed rank statistic is symmetric about its
mean. Thus, Tn and Tn are symmetric about E(T ) and
1 c 111
E(Tn ), respectively. Since Tn and Tn are independent
c 1 c
(conditional on Nj = n^ and Nc = nc), the symmetry of Tn n
follows.
To prove part d), note that
P (T
nl nc
= k) = P(L1Tn + L9 Tn = k) =
2 n,
l P(LlT1- | L2T kc)P(L2Tn k ) ,
{kc> C
where {kc}
l P(LlTn, k-kc>p {kc} 1 c
set of all possible values of E2Tn .


61
Now using the nonparametrica1ly distribution-free property
of Tn and Tn established in Lemmas 3.2.1 and 3.2.2, it
follows that for fixed L, and L0, L,T and L0T are also
1 Z in, Z n
c

nonparametrically distribution-free
can be obtained using the fact it is a convolution of two
Wilcoxon signed rank test statistics' null distributions.
Thus, for fixed and L2 the distribution can be tabled.
Tables in the Appendix 1 give the critical values for Tn n
with Lj = 1 and L2 = 1 for n^ = 1,2,.. .,15 and
n£ = 1,2,...,10 at the .01 .025 .05 and .10 levels of
significance. The actual a-levels are also reported for the
cut-offs given. The decision rule for the test is to reject
Hq if the calculated test statistic is greater than or equal
to the critical value given in the table at the desired
level of significance. A two tailed test (i.e., for Ha:
a
Oj a2) could be performed by using the symmetrical
property of the null hypothesis distribution and the table
to determine the lower critical value for the test
statistic.
A test of Hq for larger n^ and n£ can be based on the
asymptotic distribution of Tn
which will be presented in
Section 3.4


62
3.3 y^ = U2> Unknown
In the previous section, the common location parameter
was assumed to be known. Generally, this is not the case.
More often we may assume a common location parameter, but
this parameter is unknown. This section will present a
slight modification to the test statistic T_ n to be used
nl nc
in these settings. The modification will be to estimate the
common location parameter using a "smoothed" median
estimator based on the product-limit (Kaplan Meier) estimate
of the survival distribution (Kaplan and Meier, 1958). This
estimated location parameter M, replaces y in the previous
definitions. That is, define the variable to be
i = l ,2
M
M
X
1 i
The definitons of T R, y Q. T T and T
i iii n. n n , n
J J 1 c 1 c
remain unchanged. In this section, the statistic will be
denoted by TMn n to identify the fact the location
parameter was estimated with a "smoothed" median estimator
based on the product-limit estimate of the survival
distribution. This estimation does not affect the results
in Section 3.2, but Lemmas 3.2.1 and 3.2.3 c) must be
reproved, since in the proof of 3.2.1, we utilized the
independence of the Z^'s, a condition which no longer
n
c
were based on sets of


63
mutually independent observations. This is not the case in
the current context.
First, we introduce the "smoothed" median estimator and
the product-limit estimate of the survival distribution.
Let (Y(l)Y(2)-**Y(2n1+n2+n3)) represent the ordered
uncensored observations. (This ignores the fact the
original observations were bivariate pairs, and considers
only the 2n^+n2+n3 uncensored observations, i.e., 2n^
components belonging to type 1 pairs, the n2 uncensored
components of type 2 pairs and the n^ uncensored components
of type 3 pairs.) That is, X^j = if is uncensored
and X^j has rank k when ranked among the set of all
uncensored observations from either (both) components of the
pairs for i=l,2 and j=l,2,...,n. Let n(^)>
i = l 2,...,2n^+n2 + n3, be the number of censored and
/
uncensored observations which are greater than or equal to
Y(i). Thus,
n
U)
2
= I
n
l
I(X
i = lj=l
ij
Y(i)}
whe r e
I is the indicator function which takes on a value of one
when the argument is true and zero otherwise.
The product-limit estimate of the survival distribution
is defined as


64
r
s(t) =
i
j
n (n
k = l
if t < Y
(1)
ur 1)/n(k> lf Y(j) c < Y(j+n f0,r
j=l, 2 ,...,2n1+n2+n3-l
if t > Y
(2n +n2+n )
(Note, that Y^^ is the smallest uncensored observation and
^(2nj+n2+n^) t^ie larSest uncensored observation.)
The definition given here assumes no ties in the uncensored
observations. This is valid under assumptions A2 and A3.
Using the above definition, the "smoothed" median estimator
M is
M =
where
and
^ S(m ) 0.5
+ T x (m m )
S(mx) S(m2) Z 1
<
if m^ t m2
if ml = m2 ,
m1 = minCY^^: S (Y ( )) > V2 }
m2 = max{Y^. j: S(Yq^ < V2 }
A brief explanation of this estimator follows.
The product-limit estimate of the survival function,
S(t), is a right continuous step function which has jumps at


65
the uncensored observations. An intuitive estimate for the
common median is the value of Y ^ ^ such that S(Y^j) =
which often does not exist due to the nature of S(t). Thus,
the "smoothed" estimator was suggested by Miller (1981, pg .
75), which can be viewed as a linear interpolation between
m^ and If the Y ^ ^ exists, such that S(Y^)) = V2 then
m^ = m2 and M is that value of Y^.^ by definition.
Lemma 3.3.1: The statistic M is a symmetric function of the
sample observations.
Proof :
"/c "fc ^|p
Let (Y^j,Y^2)Y/2n)) represent the ordered 2n
observations where Y^^ < ^(2) ** ^(2n) This again is
ignoring the fact that the original observations consisted
of n bivariate pairs and treats the sample as if it
consisted of 2n observations (some of which are censored).
Under assumption A2, there are no ties among the uncensored
observations. Similarly, by assumptions A2 and A3, there
are no ties between an uncensored and a censored
observation, although there may be ties (of size two) among
the censored observations because type 4 pairs contribute
two components with the same value. The product-limit
estimator S(t) can be viewed as a function of the vectors
^|p ^|p ^|p
(Y(1),Y(2)* * *>Y(2n)) and (r(1)I(2) * X(2n)) where


66
(j)
1 if Y,.. is censored
(j )
0 otherwise
in the fact that
2 n
/, \ = y y i(x..> y,. )
(1) lilj-l ^
X
= 2n + 1 (rank of Y^^ in (Y(i)>^(2)> ' >Y(2n)^
In addition, S(t) can be expressed as
S(t) =
1
0
t < min Y , : I, =
1 (i) (i)
1}
t > ma x Y x : I/.N
1 (i) (i )
= 1
n
2n j
X(j)
* ^2n j + 1
V (j)
otherwise
Thus, S(t) is a symmetric function with respect to the
sample observations and therefore M, being a function of
S(t), is also. Cl
Lemma 3.3
. 2 : Conditional on n^ Tn has the same null
distribution as the Wilcoxon signed rank statistic
Proof :
Let V = {Tj, 'P 2 where = and
R = R, R R I with R. =
1 1 2 n ^ i
Let V be any arbitrary element of
P = i V is a 1 x n, vecto
1 -o -o 1
absolute rank of Z ^


67
(of which there are 2 different elements), and let r be
any arbitrary element of
R = {r : r is a permutation of the integers l,2,...,n^}.
Now, under the null hypothesis,
(X
liA2i
,C.) 2 (x
f
2 i
iCi>
t f
and thus letting = min(X^,C^) and X2^ = mi n ( X2 ^ C ^ ) it
follows that
(X2i>Xll)
for i=1,2,.
Now, let k
n^ and these pairs are also exchangeable.
}. be an operator such that
(X
li
X2 i )

if
k = 1
(X2f Xll>
if
O
II
Thus, under Hq and using the exchangeability property, it
f o 1lows
l(Xll- X21> (X12 X22>>-"'> d k1 k2 k
= {(xlr x2r ) (xlr x2r ) ,...,(xlr x2r ) ni} .
1 1
(3.3.1)
Recalling that M = the estimate of the location parameter,
is a symmetric function of the components of the observation


68
pairs from Lemma 3.3.1 and defining a function
fl(yly2) = lyl Ml ly2 Ml = Z
it follows from applying this function to (3.3.1) that
{ Z, Z , Z | Z y Z y . y Z
1 1 2 n J 1 r r r 1
1 12 n^
k k k
= {(zr ) \ (zr ) ,..., (zr ) ni}
(3.3.2)
where (Z ) =
r .
i
if k = 1
-Z if k = 0
r .
Now defining a function 2(Z) = (Y R) where 'f and R are
1 x n^ vectors such that
and
V
II
r-)
1
if Z.
J
V
o
0
k
if Z .
J
< 0
= absolute rank of Z^ ,
i e ,
rank of j Zj | among
{ | Z i | y
1 Z 2 1
> j j
Zn,l>
for j=1,2,...,n,. Applying this function to (3.3.2) it
follows that
(^ 1 ^ o > f j R > R )
12 n ^ 1 2 n ^
= (T V V R R . R )
r, r r r r r
12 n^ 1 2 n^
H kl k? k
= {(Yr ) (V r ) >...,('Fr ) "i, R r R r ,...,Rr }


69
where (f ) 1 =
i
if k.= 1
i
1 r. i
i
Now since k and r were arbitrary vectors, it follows that
P(Y = T*, R = R*) = P(T = X R = r) =
1 1
x
1
1
Thus noting this produces the same null distribution for the
Wilcoxon signed rank statistic, the proof is complete. Q
Lemma 3.3.3: Conditional on n^ and n Tn and Tn are
1 c
independent.
Proof:
This proof is done in a series of steps which are
stated as Claim 1 to Claim 7 in an attempt to avoid
confusion.
Let y £ be defined as before and let (x^,c^) denote the
. v
observed value of the iz type 2 or 3 pair
i=n^+l,...,nj+n Note, one component was censored, and
thus its observed value was c^ while the other component was
uncensored and its value is denoted by x^. This is not
specifying which component (x^ or X2i^ was censore<3.


70
Claim 1: y^ is independent of (x^,c^).
This follows by noting that under Hq and using the
exchangeability property of type 2 and 3 pairs (as was
shown in Lemma 3.2.2) that
P{yi = l | (xi,ci)} = P{X1i=xi,X2i = ci| (x ct) }
= P{ X1 i = ci X21 = xi | (xi c ) } = P{ Y^O | (xj. c )}
Since P{ Yi = l | ci ) } + P { y = 0 | ( x c ) } = 1, Claim 1
follows. Now define y = (Yn + i Yn +2>**>Yn +n
1 1 1 c
Claim 2: y is a vector of n i.i.d. Bernoulli random
variables which are independent of
^xnL + l cn1 + l ^ ^xnL + 2 cn1 + 2 ^ * (xn1 + nc cni+nc^
This follows from Claim 1 and the fact that
{ (xnj + l cn1 + l ^ ^xn:+2 cni+2 ^ * ('Xn1+nc Cn1 + n(;
)}
are i.i.d.
Claim 3: y is independent of
* ^xnj + l cn1 + l ) (xni+2 cni+2 ^ * (xn1 + n(, Cn1+n(.)
x and x where
~nl ~n4
~nl {Ull,X12)(xl2x22)>---. and
-n< = {(cn,+n +lcni+n +1) * (cncn^}
4 I C 1C
(i.e., the observed totally uncensored type 1 pairs and


71
the observed totally censored type 4 pairs,
respectively ) .
This follows from Claim 2 and the fact y is a function of
the type 2 and 3 pairs only.
Claim 4: y is independent of xn
in,
5 where x
( nc )
1 ~ 4 "c' "*c
denotes the observed ordered uncensored
members of type 2 and 3 pairs and c/n \ denotes
~ n c '
the observed ordered censored members of type 2 and 3
pairs .
Note, this claim follows directly from Claim 3 and the fact
that X/n \ and c/n \ are functions of
c' ~ c '
^^n^l cni + l ) > (xn1+2 cn1+2 ^ (xn1+n£ > cn1+nc) ^ only *
Claim 5:
yc is independent of x xQ^, x^n ^
Y
and c
( nc )
where y = {y ,y
~c c(l) c(2)
element of c^n ^ and yc
th
(i)
to the pair of which c
(i)
C(n )) c(i) ls the r
is the y which corresponds
was a member.
This claim follows from Theorem 1.3.5 of Randles and Wolfe
(1979) and since yc is a fixed permutation of y. Note that
the i.i.d. property still holds for the y 's.
(i)


72
Claim 6: Given x x X/ \
-nj ~ n4 ~(nc)
a random variable; that is
observed .
and S(nc)
Tn is no longer
the value of T.
i s
This follows directly from the definition of Tn .
Claim 7: Note, that
n + n n, + n
1 v c 1 v c d +
T I Y, Qj I J Yc W
c j=n1+l J J j=n1+l (j)
which shows that T is a function of y and is
nc 1 c
independent of x x x, \ and C/ \.
~nl -n4 ~(n ^) ~Cnc)
Thus, Tn has a null distribution equivalent to the Wilcoxon
c
signed rank null distribution and is independent of T
nl
which is a function of xn > xn > x(n ) and cjn \ only. Q
'*1 ~ 4 ~v c ^ ~ v c'
With the proof of Lemma 3.3.3, Theorem 3.2.4 is valid
for
the modified test
statistic TM .
nl ,nc
That
is, under HQ
and
conditional on n^
and n TM has
the
same
c nl > nc
distributional properties stated in Theorem 3.2.4 for T
nl nc
and the tables in the appendix are valid.


73
3.4 Asymptotic Properties
In this section, the asymptotic distribution of the
test statistic T
_ (and TM
n ^ nr ni ,n,
) under H will be
-c Jl*uc
established. The asymptotic normality of the test statistic
will be presented first, conditional on = n^ and Nc = nc
both tending to infinity and second, conditional on n
tending to infinity. In the second case, this is the
unconditional asymptotic distribution since it only requires
that the sample size go to infinity. Note that, under
assumption A.5 (A.5 stated that the probability of a type 4
pair is less than one), as n -v , N^ + Nc = (n number of
type 4 pairs) - also. The asymptotics will be presented
for the test Tn n only. In the previous section, it was
shown that under H and conditional on N, = n, and N = n
o lice
T and TM have the same null distribution; that is
nlnc nlnc
T = TM
nlnc
n n Therefore, they have the same cumulative
distribution function and thus their asymptotic
distributions are the same. There is no need to prove them
separately.
Theorem 3.4.1: Conditional on N, = n, and N = n under H
11 c c o
nlnc
- E ( T
niV
(Tn n )
nlnc
N(0,1) as n^ + and nc +
where


74
E ( T
nlnc) = (Llnl(nl+1) + L2nc(nc+1))/4
and
a(Tn n ) = [Va r(T
nlnc nln,
)}lf2
= [(L^n1(n1 + l)(2n1 + l) + L2nc ( nc + l ) ( 2 nc+l ) ) / 2 4 ]l/2
Proof:
First, it will be shown that T and T have asymptotic
nl nc
normal distributions. Without loss of generality, it will
be assumed that p = 0.
Note that
y ?. r =
. i i
i = l
I n|x2i| |xldL I) + l l V( IX211 |x1| + |x2i
i = l
where
and
2 n
1 < i < j <; n
2jI lXlj
1
n,
Dini' ~Vi-iT(|X21^'|Xi1^
! ST J. .1 T(lX2i
(,)
l', 1 < i < j < n
X1i I + lX2j I lXlj I }
1
are two U-statistics (Randles and Wolfe, 1979, page 83). It
f o 1lows
, a/2
(T
(/)
3/2
(n )
V"l+ 1)/4) E(V>)
(2) 1
+ (
"l)1/2(U2,E(U2,n>)


75
Now notice, 0 < U. <1 and under H, E(Ui ) =
i n i 1 >n ^
P { ( I X2 i I |xlih > > = 14* so that lUl,ni l,l\ < \
Theref ore ,
(n, )
3/2
(n )
3/2
(U V2) < 7 r-r *
n j v 1 n 1 z; n (n 1 )
0 as n .
(/)
(n,)V2
Thus, (T n (n.+ 1 ) / 4) and (n.)^n V2) have
(2) 1 1
the same limiting distribution as n - .
By Theorem 3.3.13 of Randles and Wolfe (1979), it is
seen that (n )^n l^j has a limiting normal distribution
1 n
o
with mean 0 and variance r E, ^ (provided £ ^ > 0)
where
r2?1 = 2 2{E[Â¥( |x2i
Xlil + IX2 j I lXljP
x ?(|X2| | x i i | + | X2 k I lXlklP ^4 >
= 1/3
Thus ,
T n.(n.+ 1)/4
nl 1 1 d
U)
+ N(0,1) .
Note that,
i ,v2
U1) W
1
n (n + 1)(2n + 1 ) ,
_ 1 j fz
P
- 1
24


76
as n1 > oo Therefore (after applying Slutsky's Theorem
(Theorem 3.2.8, Randles and Wolfe, 1979)
T n (n + 1 ) /4
1 1 1 d_
a (T )
nl
N ( 0,1 ) .
Similarly,
n, +n
1 c
n. +n
1 c
l Yj +
V
j-n,+l n ^ +1 < j T = y Y.Q.
n J J
c J =n ^ +1
l { Y j f ( c j ck) + Yk(ck
= j)}
nc(D3,n > + (2C)(U4, >
where
and
U4,n
1
n, +n
1 c
3 ri = YJ
c J =n^ +1
1
l l Y i ^ (c - c, )
fc\ n.+1 l 9 J 1 1 c
k) + Yk'i'(ck Cj)}
are two U-statistics. It follows that
"2 E_ rT ^ j. \ ^ c
n ^ n
(2C)
c(c+ DM) (o3 ECO ))
(,C)
+ )


77
Note
that 0 < n < 1 so | n V2I < V2 and thus
* r* 9 C
, ,3/2 ,3/2
(n ) (n )
c t 1 / >, c
(u, V2) < C'
n ^ 3 n v n (n
+ 0 as n *
w)
c c
1 ) c
u> 1,
Thus, (T^ n^(n^+ 1 ) / 4) and (n^'^U^ ^ V2) have
n v n c c
Uc)
the same limiting distribution as n^> <*> .
Again applying Theorem 3.3.13 of Randles and Wolfe
(1979), it is seen that (n ) U. Vo] has a limiting
c v 4 n "
c 0
normal distribution with mean 0 and variance r £ ^
2
(provided r (j>0 ), where
r2^ = 22{E[(YjVCCj Ck) + ykf(ck c.))
X (Yj'KC. Ci) + YiT(Ci Cj ) ) ] V4 }
= 22{E(Yj'i'(Cj-Ck) + YkKCk-C. )](Y.T(C.-C1) + Y^(C.-C. ) ) V4} .
By the independence of
Yj and Cj (Lemma
r2^ = 22{P(Yj =
+ p(Yj- i)p(y1=
+ P(Yk= 1 )P(Yj =
1 ) P(Cj > ck,
1)P(C > C
J <
l)P(Ck> c ,
3.2.2),
C.> C. )
J 1
c.> c.)
1 J
c.> c.)
J 1


78
+ P(yk= 1)P(Y.= l)P(Ck> Cj C.> C.) -I/4}
= 4 {V2 P ( C > C, C > C ) + Va P ( C > C. C > C )
z J k j 1 H j k 1 j
X p(ck> c., C> C.) +V4P(Ck> c., c.> C.) -1/4}
. A/I.If 1*1*11 -L .1
Thus ,
T n (n + 1 )/4
n c c ,
c d
+ N(0,1) Noting that
n t 1 }/2
(,c)
n (n + 1 ) ( 2n + 1 ) i7
c c C nVo
+ 1 as n *
c
24
and applying Slutsky's Theorem, it follows that
T -n(n+l)/4
n c c ,
-* N(0,1 ) .
o(T )
n
c
The conclusion of Theorem 3.4.1 then follows by writing
T E(T ) (L.T + L T ) (L E(T ) + LE(T ))
n, ,n n, ,n In, 2 n In, 2 n
l cl c 1 c 1 c
o(T )
nlnc
(L2 a2 ( T ) + I2 a2 ( T )),2
1 n, 2 n
1 c
L. o(T ) T E(T )
1 n ^ n n ^
La(T ) T E(T )
2 n n n
c c c
a(T n )
nlnc
o ( T )
nl
a(Tn )
nlnc
a ( T )
n
c
x
X


79
applying Slutsky's Theorem and utilizing the fact T and
Tn are independent, conditional on = n^ and N£ ~c
= n

Next, and most importantly, the unconditional
asymptotic normality of T will be established as n
n 1 n c
tends to infinity in Theorem 3.4.4. Prior to proving this,
several preliminary results will be stated which are
necessary. These preliminary results which are stated in
Lemmas 3.4.2 and 3.4.3, were proved by Popovich (1983) and
thus will be stated without proof. Minor notational changes
are made in the restatement of his results to accommodate
the notation in this dissertation.
The first preliminary result, Lemma 3.4.2, is a
generalization of Theorem 1 of Anscombe (1952).
Lemma 3.4.2: Let {T } for n,=l,2,..., n =1,2,..., be
n 1 >n c 1
any array of random variables satisfying conditions (i) and
( ii ) .
Condition (i): There exists a real number y, an array
of positive numbers {cun n } and a distribution function
F ( ) such that
lim P { T y < x a) } = F (x)
min(n^ ,n )-* 1 c 1 c
at every continuity point of F().
Condition (ii): Given any e > 0 and n > 0, there
exists v = v(e,n) and d = d(e n) such that whenever
min(n^,nc) > v, then


80
p{ T ,-T < e m for all n ,n' such that
I n n n,,n' n,,n 1 c
1 c 1 c 1 c
In' n,| < dn., |n n | 1 n .
I 1 1 I 1 1 c c' c
Let (nr} be an increasing sequence of positive integers
tending to infinity and let {N^r} and {N c r} be random
variables taking on positive integer values such that
N. p
> X as r > for some X. such that 0 l i i
n
r
i=l,c. Then at every continuity point x of ?()
lim P{Tn
r >oo 1 r
N
cr
y < xto , r , }
[Xinr],[Xcnc]
F (x)
where [a] denotes the greatest integer less than or equal to
a .
Proof :
This is Lemma 3.3.1 in Popovich (1983).
The last preliminary result necessary is a result of
Sproule (1974) which is also stated in Popovich (1983) as
Lemma 3.3.3. It can be viewed as the extension of the well
known one sample U-statistic Theorem (Hoeffding, 1948) but
with the sample size as a random variable.


81
Lemma 3.4.3: Suppose that
U
n
where B is the set of all subsets of r integers chosen
without replacement from the set of integers {l,2,...,n} and
f(t ^ t2>..., t ) is some function symmetric in its r
arguments. This Un is a U-statistic of degree r with a
symmetric kernel f(). Let {nr} be an increasing sequence
of positive integers tending to infinity as r > and (Nr)
be a sequence of random variables taking on positive integer
2
values with probability one. If E{f(X^, X2.,X )} <
1/ o Nr P
lim VarCn^ U ) = r ? > 0, and 1 then
n 1 n
n+ r
r \ r2^)^ } = Mx) ,
lim P{(UN E(Un )) < N
r-)- r r
where $ ( ) represents the c.d.f. of a standard normal random
variable.
Proof: This is Lemma 3.3.3 in Popovich (1983). ^
One comment is needed about this result. The proof of this
lemma follows as a result of verifying that conditions
and C2 of Anscombe (1952) are valid and applying Theorem 1
of Anscombe (1952). Condition is valid under the null
hypothesis and the verification of condition C2 is contained
in the proof of Theorem 6 by Sproule (1974). This condition


82
C 2 will be utilized in the proof of the major theorem of
this section which follows.
Theorem 3.4.4: Under Hq ,
\ ,Nc E(TN, ,N
a(TN,,N }
1 c
N(0,1) as n .
Proof:
The proof which follows is very similar to the proof of
Theorem 3.3.4 in Popovich (1983).
Let T
nlnc
T E (T )
nl ,nc nl ,nc
o(T )
nlnc
, the standardized
T statistic. Theorem 3.4.1 shows that {T } for
n1nc nlnc
n^=l,2,..., n =1,2,..., satisfies condition (i) of Lemma
3.4.2 with y = 0 amd A5, it can be seen that X^ > 0 for at least one i=l,c. If
X^ = 0, for i=l or i=c, then Theorem 3.4.4 follows directly
from Theorem 1 of Anscombe (1952) and Lemma 3.4.3. Thus, it
will be assumed that X ^ > 0 for i=l,c. The proof of
Theorem 3.4.4 follows if it can be shown that condition (ii)
of Theorem 3.4.2 is satisfied.
Let T
: E (T )
nl ni
a (T )
nl
the standardized T
n,
statistic
In the proof of Theorem 3.4.1, it was shown that


83
T has a limiting standard normal distribution by utilizing
nl
the U-statistic representation of T
1
As a result of Lemma
3.4.3 and this U-statistic representation, it follows that
£
T satisfies condition C., of Anscombe ( 1 952) (since T is
n 1 L nl
equivalent to a U-statistic which satisfies condition C2 of
Anscombe (1952) as proved by Sproule (1974)). This
condition C2 can be stated as follows.
Condition C2: for a given e^> 0 and n > 0, there
exists Vj and d^> 0 such that for any n^>
* *
T T ,
n 1 nl
< e for all nj such that |n|
- n,
< d1n1 } >
1 n
(3.4.1)
Similarly, as a result of the U-statistic representation of
Tn (as shown in the proof of Theorem 3.4.1) and from Lemma
c
3.4.3, it follows that T
T E (T )
n n
c a (T )
n
c
satisfies
condition C2 of Anscombe (1952). That is, for a
given z0 and n > 0, there exists V2 and d2> 0 such that
for any n^>
P{ T T < e for all n such that n
in n I 2 c I c
c c
n < dn } >
cl 2 c
1 n
(3.4.2)


84
Consider
T E(T
nlnc
nlnc
nlnc
a(Tn n }
nlnc
L.o(T )
1 nl
a ( T )
nlnc
* V(Tn >
) + *
1
a (T )
nlnc
f >
K )
Note that,
(1) L^n and L2n ate functions only of N^ and Nc and the
given and constants.
(2) (Lln)2 + (L2n)2 = 1
I ! P I
(3) There exists constants and 1^ such that * L^
. P .
and L * L as n + .
7 n 7
First, it will be shown that condition (ii) is
f £ f -Jc
satisfied for L.T 1 + L_T ) = T
1 ^ n, ; 2 ^ n -1 n , n
1 c 1 c
Let e > 0 and n > 0 be given and let v^, v2, ^2
satisify (3.4.1) and (3.4.2). Let v = raax(vp v2) and
d = min(d^, d2). Now,
P { T , T < 2 e for all n1, n such that
In' n' n, ,n I 1 c
1 c 1 c
< dn. ,i=l,c)
i 1 i
1 *
* i
' i *
* 1 "V
T
- T 1
+ L J T ,
- t h
1 n'
n, 1
2 1 n '
n 1 1
< 2 e for all n', n'
1 c


35
such
that | n

i
n.
i 1
< dn
i 1 =
1 ,c}
*
V -
1
* i
T i
nl
< t and
t .
l2I
*
V
c
* i
" T j
n i
c
< £
for
all
S
such
that | n
t _
i
n .
i i
< dn
i 1 =
1 c}
*
* 1
T ¡
nl
< e for
all
ni
such
that
lni
' nl
<
*
' n "
c
* i
T
n i
c
< e for
all
n
c
such
that
In
i c
- n
c
| <
i *
lTn:
*
- T
n,
¡ < e or
I
l2I
*
T ,
n
*
- T
n
1 ^ E
for
all
S
c c
such that n' n, < dn, and n' n < dn }
11 II 1 I c cl c
f £ £ i i i
> P{L. T T < e for all n' such that n' n. <
II n j n^l 1 II II
1 I * I I I
+ P{L_ T T < e for all n' such that n' n <
2 I n 1 n
c c
c c
(3
Now using inequalities (3.4.1) and (3.4.2) and applying
to (3.4.3) with e = min{e^(L^) ^ ^ } then
P T . T < 2e for all n,, n such that
1 I n, ,n' n,,nl 1 c
1 c 1 c
ni ~ nil < dni> 1=1>c} > (1 n) + (1 n) -1 = 1 2n
n*
c
dn l }
dn }
c
n'
c
dn l }
dn } 1 .
c
.4.3)
them


86
Therefore T satisifies condition (ii) of Lemma 3.4.2 so
nl *nc
that Theorem 3.4.4 is valid for T_ = L,T + L0T.
nl nc
In. 2 n '
i c
To see that the Theorem is valid if and L2 are replaced
t I
^ and respectively, consider,
nl >nc
Tninc Lln Tnl
c
' *
- LlTn
' *
2 n n,
+
' *
L2Tn,
= (L
1 n
- + (3.4.4)
Now, since Tn and TQ converge in distribution to standard
normal random variables, Tn and Tn are 0p(l) (Serfling,
1 n c
I ^ T
( 1980 ), pg. 8). Also, since L^n > L ^ and
t P 11 11
L 2 n + L2 as n > , (Lln L ^) and (L2n L2) are op(l).
Therefore (3.4.4) shows that
(L^n L^)T* + (^2n ^2^n p ( 1) and thus, Theorem
3.4.4 is valid.

3.5 Comme n t s
From the results in Sections 3.2, 3.3,and 3.4, it is
clear that a distribution-free test of the null hypothesis
of bivariate symmetry versus the alternatives presented
could be based on T (or TM ). For small samples,
nlnc nl>nc
an exact test utilizing the distribution of T (and
n 1 ,nc
TM
) conditional on N, = n,
,nQ' 1 1
and N_ = n could be


87
performed. For larger samples, the asymptotic normality of
T (and TM_ ) could be used. In Chapter Five, a
n1nc nlnc
Monte Carlo study will be presented which compares the CD
test with the two tests presented in this chapter. For
each, the asymptotic distribution will be used for samples
of size 25 and 40 to investigate how the statistics compare
under the null and alternative hypotheses for various
distributions. First though, we make some comments on this
chapter.
Comment 1
In Section 3.2, the test statistic Tn n conditional
on n£, was presented which had a null distribution
equivalent to the Wilcoxon signed rank statistic. If
instead of conditioning on nc, the statistic had been
presented (with some minor adjustments) conditional on n2
and n^, the statistic would then have had a null
distribution equivalent to the Wilcoxon rank sum
statistic. Conditioning on n£ and not on n2 and n^ was
chosen because the observation of a particular n2 and n^ in
itself, seemed important. That is, if only type 3 pairs had
occurred (ignoring the number of type 4 pairs) that was
significant, since under the null hypothesis, the
probability a bivariate pair is type 3 is equal to the
probabiltiy the pair is type 2. The signed rank statistic
incorporates this idea and thus was used.


88
Comment 2
In Section 3.3, the Kaplan-Meier estimate of the
survival distribution was used in estimating the common
location parameter. The usual median estimator (the sample
median) could not be used, because in the presence of right
censoring this estimator is negatively biased. Thus, the
"smoothed" estimator based on the Kaplan-Meier estimate of
the survival distribution was the logical choice.
Comment 3
The tests presented in this chapter are not recommended
for situations in which heavy censoring occurs early on,
that is, a lot of censoring in the smaller measurements. If
this heavy censoring was to occur, many type 4 pairs would
be present in the sample which are not used in the
calculation of the test statistic other than to estimate the
common location parameter. This test was more designed for
situations when the extreme values (i.e., the larger values)
tended to get censored.
Comment 4
In this chapter, statistics were presented to test for
differences in scale when (1) the common location parameter


89
was known or (2) the common location parameter was
unknown. The next natural extension would be to test the
null hypothesis of bivariate symmetry versus the alternative
that differences in scale existed with unknown location
parameters which could be potentially different. This idea
could be incorporated into the test statistic by using
separate "smoothed" estimators for and This idea
will be further investigated in Chapter Four.


CHAPTER FOUR
A TEST FOR BIVARIATE SYMMETRY VERSUS
LOCATION/SCALE ALTERNATIVES
4.1 Introduction
In Chapters Two and Three test statistics were
presented to test the null hypothesis of bivariate symmetry
versus the alternative hypothesis that the marginal
distributions differed in their scale parameter. This
chapter will consider a test for the more general
alternative, that is, that the marginal distributions differ
in location and/or scale. To do this, two statistics will
be made the components of a 2-vector, W of test
statistics. The first statistic denoted TE_ is a
n 1 nc
statistic which is used to detect location differences. It
was introduced by Popovich (1983) and is somewhat similar to
the statistic introduced in Chapter Three. The second
component of the 2-vector will be a statistic(s) which is
designed to test for scale differences. Three different
statistics will be considered for this second component.
They are (1) TM (Chapter Three, Section 3.3), (2)
nlnc
TM but using separate location estimates for X and
n 1 y n c ^
X2^ and (3) the CD statistic (Chapter Two). It will be
shown in Sections 4.2 and 4.3 that the distribution of Wn is
90


on
not distribution free, even when H is true. Thus, if f is
o rn
the variance-covariance of W the quadratric form W'
~ n n ~nrn~n
will not be distribution-free. A consistent estimator of
A
j] n j-n t>e introduced in Section 4.5 and a test based
the asymptotic distribution-free statistic U' i W will be
recommended for large sample sizes. For small sample sizes
a permutation test will be recommended. First though, we
introduce the TE statistic by Popovich (1983) with a slight
change in notation to accommodate this thesis.
Let
Di
= Xli
- x2i
a nd
R( |
Di|)
be
the
absolute
Di
for i
= 1,2
> >
n, that
is ,
R(
1 Di 1
is
the
rank of |
among (|
Dll>
1 ^2 1
|Dn
| ).
Define
1
if
Z .
^ 0
T .
1
= ¥(D1)
= <
1
0
if
Z .
1
O
V

Let
TE
nl
a nd
TEn
c
be defined
to
be
t he
following:
nl
TE
nl
i
II 0^1
H-*
4'1 R (
lDi
)
and
TEn = N N2 .
c
Notice that TE
is the Wilcoxon signed rank statistic
applied to the n^ totally uncensored pairs. Popovich (1983)
showed under Hq, N^ is distributed as a Binomial random
variable with parameters nc and p = V2 ?2(0) = V2 P(type 2 or
3 pair). With a slight modification from Popovich, the
n 1 n<
statistic TE
is


92
where
+ K2n
TE
TE n. (n, + 1 )/4
n ^ 1 1
1 (r^Ct^ + 1 ) ( 2n L + l)/24)1/2
and
TE
TE
"c (
(n ) z
c
and K^n and K2n are a sequence of random variables
satisfying:
1) and I<2n are only functions of and Nc,
2) there exists finite constants and such
P P
that Kjn + Kj and ^2n K2 as n .
This is slightly different from the statistic Popovich
introduced, the difference being that he required
Kjn = (l~K2n) which is not being required here.
One comment before proceeding to Section 4.2. In this
Chapter, type 4 pairs will be ignored (except in estimating
the location parameter for the scale statistics). This has
no real affect since TE_ n > TMn and CD are not
n1 nc nl c
affected by their presence (other than in estimating the
location parameter). It will be assumed that the sample is
of size n


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            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            Wayne [C. Huber
            Professor of Environmental
            Engineering
            This dissertation was submitted to the Graduate Faculty
            of the Department of Statistics in the College of Liberal
            Arts and Sciences and to the Graduate School, and was
            accepted for partial fulfillment of the requirements of the
            degree of Doctor of Philosophy.
            Augus t, 1984
            Dean for Graduate Studies
            and Research


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            BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
            BY
            LAURA LYNN PERKINS
            A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
            OF THE UNIVERSITY OF FLORIDA IN
            PARTIAL FULFILLMENT OF THE REQUIREMENTS
            FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
            UNIVERSITY OF FLORIDA
            1984

            to my parents, with
            love

            ACKNOWLEDGEMENTS
            I would like to thank Dr. Ronald Randles for originally
            proposing the problem. Without his enormous patience,
            encouragement and guidence, it would not have been
            possible. I would also like to thank Dr. Jim Kepner for his
            help in its original conception. To my family, especially
            my parents, I am grateful for the mental and financial
            support they provided when I needed it the most. I would
            like to thank Robert Bell for his patience and
            understanding. More than once, when I could not see the
            end, he was there to reassure me and give me confidence. To
            my typist, Brenda Prine, I express my gratitude for many
            hours spent with no complaints. Last, but not least, I
            would like to say thank you to the Department of Statistics
            for making this all possible.
            iii

            TABLE OF CONTENTS
            PAGE
            ACKNOWLEDGEMENTS iii
            ABSTRACT vi
            CHAPTER
            ONE INTRODUCTION 1
            TWO A STATISTIC FOR TESTING FOR DIFFERENCES
            IN SCALE 16
            2.1 Introduction . 16
            2.2 The CD Statistic 19
            2.3 Permutation Test 35
            2.4 Asymptotic Results 39
            2 .5 Comments 44
            THREE A CLASS OF STATISTICS FOR TESTING FOR
            DIFFERENCES IN SCALE 48
            3.1 Introduction 48
            3.2 y i = u 2 Known 50
            3.3 p^ = P 2 Unknown 62
            3.4 Asymptotic Properties 73
            3 .5 Comment s 86
            FOUR A TEST FOR BIVARIATE SYMMETRY VERSUS
            LOCATION/SCALE ALTERNATIVES 90
            4.1 Introduction 90
            4.2 The W Statistic Using
            4.3 The Wn Statistic Using
            CD Ill
            4.4Permutation Test 121
            4.5Estimating the Covariance 123
            IV

            FIVE MONTE CARLO RESULTS AND CONCLUSION. . . . 133
            5.1 Introduction 133
            5.2 Monte Carlo for the Scale Test . . 134
            5.3 Monte Carlo for the Location/
            Scale Test 142
            APPENDICES
            1 TABLES OF CRITICAL VALUES FOR TESTING
            FOR DIFFERENCES IN SCALE 158
            2 THE MONTE CARLO PROGRAM 171
            BIBLIOGRAPHY 183
            BIOGRAPHICAL SKETCH 185
            v

            Abstract of Dissertation Presented to the Graduate School
            of the University of Florida in Partial Fulfillment of the
            Requirements for the Degree of Doctor of Philosophy
            BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
            By
            Laura Lynn Perkins
            Augus t, 19 8 4
            Chairman: Dr. Ronald H. Randles
            Major Department: Statistics
            Statistics are proposed for testing the null hypothesis
            of bivariate symmetry with censored matched pairs. The two
            types of alternatives considered are (1) the marginal
            /
            distributions have a common location parameter (either known
            or unknown) and differ only in their scale parameters and
            (2) the marginal distributions differ in their locations
            and/or scales. For the first alternative, two types of
            statistics are proposed. The first is a statistic based on
            Kendall's tau modified for censored data, while the second
            type is a class of statistics consisting of linear
            combinations of two statistics. Conditional on N^, the
            number of pairs in which both members are uncensored, and
            N2 , the number of pairs in which exactly one member is
            censored, the two statistics used in the linear combination
            are independent and each has a null distribution equivalent
            vi

            to that of a Wilcoxon signed rank statistic. Thus, any
            member in the class can be used to provide an exact test
            which is distribution-free for the null hypothesis. The
            statistic based on Kendall's tau is not distribution-free
            for small sample sizes and thus, a permutation test based on
            the statistic is recommended in these cases. For large
            samples, a modified version of the Kendall's tau statistic
            is shown to be asymptotically distri bution-free .
            For the second and more general alternative, a small
            sample permutation test is proposed based on the quadratic
            form Wn = T^ j; ^ T , where T' is a 2-vector of statistics
            composed of a statistic designed to detect location
            differences and a statistic designed to detect scale
            differences and | is the variance-covariance matrix for
            Tn> For large samples, a distribution-free approximation
            for T' 1 T is recommended.
            ~ n T ~ n
            Monte Carlo results are presented which compare the two
            types of statistics for detecting alternative (1), for
            sample sizes of 25 and 40. Quadratic form statistics Wn
            using different scale statistic components are also compared
            in a simulation study for samples of size 35. For the
            alternative involving scale differences only, the statistic
            based on Kendall's tau performed best overall but requires a
            computer to do the calculations for moderate sample sizes.
            For the more general alternative of location and/or scale
            differences, the quadratic form using the scale statistic
            based on Kendall's tau performed the best overall.
            vii

            CHAPTER ONE
            INTRODUCTION
            Let Wj and denote random variables; then the
            property of bivariate symmetry can be defined as the
            property such that (W ^ , W 2 ) has the same distribution as
            (W2*W^). This property of bivariate symmetry is also
            referred to as exchangeability (or bivariate
            exchangeability). Commonly, this property arises as the
            null hypothesis in settings in which a researcher has paired
            observations, such as, when the subjects or sampling units
            function both as the treatment group and the control group
            or possibly the researcher has matched the subjects
            according to some criteria such as age and sex.
            For example, a dentist may want to assess the
            effectiveness of a dentifrice in reducing dental
            sensitivity. The dentist randomly selects n patients and
            schedules two appointments for each patient at three month
            intervals. During the first visit, a hygienist assesses the
            patient's dental sensitivity after which the patient is
            given the dentifrice by the dentist. At the end of the
            three month usage period, the patient returns and his or her
            f »
            dental sensitivity is again assessed. If X^ and X2^ are
            the first and second sensitivity measurements, respectively,
            1

            2
            t" Vi
            of the i 1 patient, the dentist has n bivariate pairs in the
            sample. If there is no treatment effect, then effectively
            the two observations of dental sensitivity are two
            measurements of exactly the same characteristic at two
            randomly chosen points of time. In which case, the
            ft t »
            distribution of is t^ie same as that of (X2i,X^i),
            and so a test using the null hypothesis of bivariate
            symmetry would be appropriate.
            The possible alternatives for a test which uses a null
            hypothesis of bivariate symmetry are numerous. The three
            types of alternatives which will be considered in this work
            are the following:
            1) The marginal distributions have a common
            known location parameter and differ only in
            their scale parameters.
            2) The marginal distributions have a common
            unknown location parameter and differ only in
            their scale parameters.
            3) The marginal distributions differ in their
            location and/or scale parameters.
            The situation under consideration in this work is
            further complicated by the possibility of censoring.
            Censoring occurs whenever the measurement of interest is not

            3
            observable due to a variety of possible reasons. The most
            common situation is when the measurement is the time to
            "failure" (i.e., death, the time until a drug becomes
            effective, the length of time a drug remains effective,
            etc.) for an experimental unit subjected to a specific
            treatment. If at the end of the experiment, the
            experimental unit still has not "failed," then the
            corresponding time to "failure" (referred to as survival
            time) is censored. All that is known, is that the survival
            time is longer than the observation time for that unit and
            thus has been right censored. An example of censoring in
            bivariate pairs could be the times to failure of the left
            and right kidneys or the times to cancer detection in the
            left and right breasts (Miller, 1981).
            Many different types of right censoring exist (Type I,
            Type II and random right censoring), each determined by
            restrictions placed on the experiment. Type I censoring
            occurs if the observation time for each experimental unit is
            preassigned some fixed length T. Thus, if the survival time
            for a unit is larger than T, it is right censored. Type II
            censoring occurs when the experiment is designed to be
            terminated as soon as the r*"^1 (r occurs. Random right censoring is a generalization of Type
            I censoring, in which the experimental units each have their
            own length of observation (which are not necessarily the
            same). This would occur, for example, if the length of the
            experiment was fixed but random entry into the experiment

            4
            was allowed. It is this latter type of censoring which this
            work addresses.
            Now we statistically formulate the problem of
            I »
            interest. Let (X^,X2^) f°r i=1>2,...,n denote a random
            sample of bivariate pairs which are independent and
            identically distributed (i.i.d.) and Ci i=l,2,...,n denote a
            random sample of censoring times which are i.i.d., such that
            C ^ denotes the value of the censoring variables associated
            t I
            with pair (X^,X2i). -*-n t^e case random right censoring,
            the observed sample consists of (^ii>^2i’^i) w^ere
            t T
            Xi^ = min(X,£,C^), = mi n ( ^ > C ¿ ) and 6i is a random
            variable which indicates what type of censoring occurred,
            r
            0i Description
            1
            2
            3
            4
            Xli x;ici
            Xli>Ci> x2i Xu>ci,x;i>ci
            Now we state a set of assumptions which are referred to
            later .
            Assumptions:
            I I
            Al. (^ii>^2i) i=l,2,...,n are i.i.d. as the
            t »
            bivariate random variable
            A 2
            f t
            (X^j,X2j) has an absolutely continuous bivariate
            distribution function F(
            X1 ”
            X2 ' U2
            )

            5
            O
            where F(u,v) = F(v,u) for every (u,v) in R . The
            parameters y ^ (p 2) and (a 2) ate location and
            scale paratmeters, respectively. They are not
            necessarily the mean and standard deviation of the
            marginal distributions.
            A3. Cj,C2»»..,C are i.i.d. continuous random
            variables, with continuous distribution function
            G(c) .
            A4. The censoring random variable , is independent
            f I
            of i=l»2 n and the value of is
            the same for both members of a given pair.
            A 5. P(xJi>C.,X2i>Ci) < 1.
            A6. G(F ( V2 )) < 1 where F^ denotes the marginal
            i i
            T
            cumulative distribution function (c.d.f.) of X^
            i=l,2.
            Note that under A5, the probability is positive that the
            sample will contain observations that are not doubly
            censored.
            With this notation, the null and alternative hypotheses
            can now be formally stated. The null hypothesis is
            Hq : y^=P2> °1=ct2 versus the alternatives:

            6
            1. The case where p^=y2=M with p known,
            Ha: °1 * a2
            2. The case where pj=p2=U with p unknown,
            Ha: °i * a2
            3. Ha : p^ ^ \i 2 and/or ^ 02-
            Chapter Two and Three will present test statistics for
            alternatives 1) and 2). Chapter Four will present a test
            for the more general alternative stated in 3). Monte Carlo
            results and conclusions will be presented in Chapter Five.
            First though, we describe related work in the literature.
            Since this dissertation combines two areas of previous
            development, that is, bivariate symmetry and censoring, the
            first part of the review will deal with related works in
            bivariate symmetry without a censoring random variable
            considered. The second part of the review will mention
            related works for censored matched pairs.
            The first four articles to be considered, Sen (1967),
            Bell and Haller (1969), Hollander (1971) and Kepner (1979),
            all suggest tests directed towards specific alternatives to
            the null hypothesis of bivariate symmetry. The work of
            Kepner (1979) more directly influenced the development of
            this thesis than the others, but they were direct influences
            on the work of Kepner and thus will be mentioned.
            Sen's article (1967) dealt with the construction of

            7
            conditionally distribution-free nonparametric tests for the
            null hypothesis of bivariate symmetry versus alternatives
            that the marginal distributions differed only in location,
            or that the marginal distribution differed only in scale, or
            that the marginal distributions differed in both location
            and scale. The basic idea behind his tests is the
            I I
            following. Under Hq, the pairs (xii>X2i^ i=l»2,...,n are a
            random sample from an exchangeable continuous
            distribution. He pools all the elements into one sample (of
            size N=2n), ignoring the fact the original observations were
            bivariate pairs and then ranks this combined sample. From
            this, Sen obtains what he refers to as the rank matrix,
            v / R11
            R 1 2
            R1 n
            RN "
            \ R 2 1
            R 2 2 * • *
            R2n
            where R^^ is the rank of X^^ in the pooled sample j=l,2
            i=l,2,...,n. Let S(R^) be the set of all rank matrices that
            can be obtained from R^ by permuting within the same column
            of R^ for one or more columns. Under HQ, each of the 2n
            elements of S(R^) is equally likely and thus, if Tn is a
            statistic with a probability distribution (given S(RN) and
            H ) which depends only on the 2n equally likely permutations
            of Rn, Tn is conditionally distri bution-free (conditional on
            the given R^ and thus S(R^) observed). Sen's statistic Tn
            can be defined as
            T
            n
            n
            n
            l
            i = l
            ,R
            1 i

            8
            where ^ is a score function based on N=2n and i alone.
            For the test of location differences only, Sen suggests
            using the Wilcoxon scores (E^ ^ ) or the quantile F
            scores (E^ ^ = F ^(^y) where F is an appropriately chosen
            absolutely continuous c.d.f.). The Ansari-Brad1ey scores
            (En ^ ^— ~ |i ~ —j—|) or the Mood Scores
            (Ew . = ( * - 4 )^) are suggested for use when the
            alternative is that the marginal distributions differ only
            in their scale parameters. For the more general
            alternative, that the marginal distributions differ in
            location and scale, he recommends making a vector (of size
            2) of his statistics where one component is one of the
            statistics for differences in location and the other for
            scale.
            One basic weakness of Sen's proposals, as mentioned by
            Kepner (1979), is that the procedure basically ignores the
            correlation structure within the original observations
            » t
            ^Xli,X2i^ i=l,2,...,n and, thus, suggests that a better test
            could possibly be constructed by exploiting the natural
            pairing of the observations.
            The test proposed by Bell and Haller (1969) does
            exploit this natural pairing of the observations. They
            suggest both parametric and nonparametric tests for
            bivariate symmetry. In the normal case, they form the
            likelihood ratio test for the transformed observations
            (Yli,Y2i) where Yii= xn “ x2i and Y2i= Xli + X2i* The

            9
            resulting test they suggest when dealing with a bivariate
            normal distribution is to reject Hq if |B ^ j > t(3^;n-2) or
            j B 2| > t(g25n-l) where
            (n-2)ly2 r(Y ,Y.) n^ Y
            B. = — and B „ =
            1 2 l/„ 2 c
            (l-rZ(Y1,Y2))/2 b
            and r(Y^,Y2) is the sample correlation coefficient of the
            — 2
            Yj^'s and ^2i's’ ^ j and S are the sample mean and unbiased
            sample variance, respectively of the Y^'s and t(3;n)
            represents the critical value for a t distribution with n
            degrees of freedom which cuts off 3 area in the right
            tail. The main problem with this test, as Kepner (1979)
            also states, is that the overall level of the test, a, is
            a = 23 1 + 2 3 2 - 43^
            so relatively small values for 3^ and 32 would need to be
            chosen .
            The nonparametric tests they suggest are either
            complicated, due to many estimation problems involved, or
            have low power or are just unappealing due to the fact the
            test is somewhat researcher dependent. (That is different
            researchers working independently with the same data could
            reach different conclusions.) Thus, they will not be
            mentioned.
            Hollander (1971) introduced a nonparametric test for
            the null hypothesis of bivariate symmetry which is generally
            appealing and consistent against a wide class of
            alternatives. He suggested

            10
            D
            n
            /[ lF„(x-y)
            - Fn(y,x)}2dFn(x,y)
            where
            is the bivariate empirical c.d.f. He notes that nDn is not
            distribution-free nor asymptotically distribution-free when
            Hq is true, and thus proposed a conditional test in which
            the conditioning process is based on the 2n data points
            (j j )
            {((xH»x2i)
            V 0 or 1
            for k = 1,2,...,n}
            > • •
            which are equally likely under HQ . Here we let
            (s,t)^) = (s,t) and (s,t)^^ = (t,s). This statistic
            performs well even for extremely small sample sizes (n=5)
            with one major drawback as mentioned by Hollander which is
            the computer time which it takes to evaluate nDn. It
            becomes very prohibitive for even moderate n. Koziol (1979)
            developed the critical values for nDn for large sample
            sizes, which work much better than the large sample critical
            value approximations originally suggested by Hollander.
            Kepner (1979) proposed tests based on the transformed
            observations (Y^,Y2¿) of Bell and Haller for the null
            hypothesis of bivariate symmetry versus the alternatives
            that the marginal distributions differ in scale or that the
            marginal distributions differ in location and/or scale. For
            the alternative of differences in scale, he proposed a test

            where
            1 if t>0
            'i'(t)
            0 if t < 0 ,
            which is Kendall's Tau applied to the transformed
            observations. He noted that tt is neither distribution-free
            nor asymptotically distribution-free in this setting and
            thus recommended a permutation test which is conditionally
            distribution-free based on tt for small samples. This
            permutation test was based on conditioning on what he called
            /
            the collection matrix, Cn,
            He noted that under HQ and conditional on Cn, there are 2n
            equally likely transformed samples possible,
            each being determined by a different collection of T* 's
            j|f
            where ¥^ = {1 or -1}. For larger samples, he obtains the
            asymptotic distribution which can be used to approximate the
            permutation test.
            One nice property of the statistic tt , which Kepner
            notes, is that tt is insensitive to unequal marginal

            1 2
            locations and thus location differences do not influence the
            performance of the test.
            For the more general alternative of location and/or
            scale differences, a small sample permutation test for
            bivariate symmetry was proposed based on the quadratic form
            whe r e
            T
            n
            W + is the Wilcoxon signed rank test statistic calculated on
            the Yj^'s and tt n is as previously defined. Again, the
            conditioning of the test is on the collection matrix Cn> He
            /
            obtains the limiting distribution of the small sample
            permutation test and proposes a large sample
            distribution-free approximation which is computationally
            efficient.
            The second collection of articles which will be
            mentioned deals with the topic of censored matched pairs.
            Much work has been done recently in the area of censored
            data, but the work of Woolson and Lachenbruch (1980) and
            Popovich (1983) most directly influence the results in this
            thesis and thus will be described here.
            Woolson and Lachenbruch (1980) considered the problem
            of testing for differences in location using censored

            13
            matched pair data. The situation they considered is
            identical to the situation developed in this thesis if one
            assumes equality of the scale parameters. They utilized the
            concept of the generalized rank, vector introduced by
            Kalbfleisch and Prentice (1973) to develop tests by
            imitating the derivation of the locally most powerful (LMP)
            rank test in the uncensored case. Although they imitate the
            development of LMP rank tests for the uncensored case, it is
            unclear whether these tests are LMP in the censored case.
            Scores for the test are derived for (1) if the underlying
            distribution the differences (i.e., - X2^) is logistic
            and (2) if the underlying distribution for the differences
            is double exponential. In each case the statistic developed
            reduces to usual statistic (Wilcoxon signed rank statistic
            and sign test statistic for an underlying logistic density
            or double exponential density, respectively) when no
            censoring is present. Asymptotic results for the tests are
            derived based on the number of censored and uncensored
            observations tending to infinity simultaneously.
            Popovich (1983) proposed a class of statistics for the
            problem of testing for differences in location using
            censored matched pair data. The class consists of linear
            combinations of two statistics which are independent given
            Nj and N2 where is the number of pairs in which both
            members are uncensored and N2, the number of pairs in which
            exactly one member is censored. The class of statistics can
            be expressed in the general form of

            14
            T
            n
            (N1 ,N2 )
            (l-L )V2
            T, (N. )
            In 1
            Vo *
            I/2 T„ (N.)
            n 2 n 2
            where T^n is the standardized Wilcoxon signed rank statistic
            calculated on the uncensored pairs, and
            T2n = N2_ /2
            ^N2R N2L^
            where
            N2R
            i s
            the number of pairs for
            which Xis
            censored
            and X2^
            is
            not
            , and N2L i-s the number
            of pairs for which X2^ is censored and is not (note
            ^2R+^2L= ^2^* The weight Ln is a function of and ^ only
            P *
            such that 0 distri bution-free statistic calculated only on the
            uncensored pairs (and is a common statistic used for testing
            £
            for location in the uncensored case) while T2n is a
            statistic based only on the type 2 and 3 pairs (as
            previously defined in this introduction). The statistic
            k
            T2n is designed to detect whether type 2 pairs are occurring
            more often (or less often) than should be under the null
            hypothesis. Under Hq , T2n is a standardized Binomial random
            variable with parameters N2=n2 and p= V2 and thus
            distribution-free. Popovich obtains asymptotic normality
            for the statistic Tn(N^,N2) under the conditions (1) that
            and N2 tend to infinity simultaneously and (2) under a more
            general condition as n tends to infinity. In a Monte Carlo
            study, he compares five statistics from this class to the
            test statistic of Woolson and Lachenbruch (T^) (1980) based
            on logistic scores. The results show that these statistics
            perform as well as T WL (better in some cases) and that they

            1 5
            are computationally much easier to calculate. Furthermore,
            exact tables can be generated for any member of the class
            proposed by Popovich.
            With the background established for the research in
            this thesis, the attention will now be focused toward the
            development of the test statistics to be investigated
            here. Chapter Two will present a statistic for testing for
            differences in scale which can be viewed as an extension of
            Kepner's tt n for censored data. In Chapter Three, another
            statistic will be presented for the same alternative but
            more in the spirit of the work proposed by Popovich, that
            is, the linear combination of two statistics which are
            conditionally independent (conditioned on the number of type
            1 and (type 2 + type 3) pairs observed). For the more
            general alternative (i.e., differences in location and/or
            scale), Chapter Four will present a statistic(s) which is a
            vector of two statistics (one for scale and one for
            location) following the work of Kepner. Lastly, Chapter
            Five will present a Monte Carlo study of the statistics
            developed in this dissertation.

            CHAPTER TWO
            A STATISTIC FOR TESTING FOR DIFFERENCES IN SCALE
            2.1 Introduction
            In this chapter a statistic will be presented for
            testing the null hypothesis of bivariate symmetry in the
            presence of random right censoring. Figure 1 represents a
            possible contour of an absolutely continuous distribution of
            this form. The alternative hypothesis for which this test
            statistic is developed is : o^ t i.e., the marginal
            distributions differ in their scale parameters. The
            marginal distributions are assumed to have the same location
            parameter. Figure 2 represents a possible contour of an
            absolutely continuous distribution of this form.
            The basic idea for this statistic was introduced in a
            dissertation by Kepner (1979). He suggested the use of
            Kendall's tau on an orthogonal transformation of the
            original random variables to test for differences in scale
            in the marginal distributions. The presence of a censoring
            random variable was not included. To extend this idea to
            include the presence of random right censoring, the concept
            of concordance and discordance in the presence of censoring
            which was used by Oakes (1982) was applied.
            16

            17
            Figure 1. Contour of an Absolutely Continuous Distribution
            That Has Equal Marginal Locations and Equal
            Marginal Scales.

            18
            Figure 2. Contour of an Absolutely Continuous Distribution
            That Has Equal Marginal Locations and Unequal
            Marginal Scales.

            19
            Section 2.2 will present the test statistic and the
            notation necessary for its presentation. A small sample
            test will be discussed in Section 2.3. Section 2.4 will
            investigate the asymptotic properties of the test statistic,
            with comments on the statistic following in Section 2.5.
            2.2 The CD Statistic
            In this section, the test statistic will be presented
            which is designed to test whether the marginal distributions
            differ in their scale parameters. First, since the work, is
            so related, the test statistic which Kepner (1979) proposed
            to test for unequal marginal scales will be presented. This
            will give the reader an understanding of the motivation for
            the test statistic.
            » »
            Let for i=l,2,...,n denote independent
            identically distributed (i.i.d.) bivariate random variables
            » »
            which are distributed as (X,j,X2^)* Consider the following
            t I
            orthogonal transformation of the random variables (X^,X2^);
            let
            Yli = Xli + X2i and Y2i = Xli “ X2i for
            Figure 3 illustrates what happens to the contour given in
            Figure 1 (i.e., the contour of an absolutely continuous
            distribution under HQ) when this transformation is

            20
            applied. Figure 4 shows what happens to the contour given
            in Figure 2 (i.e., under H ) when this transformation is
            applied. Note, as can be seen in Figure 3, under this
            I »
            transformation and H , Y, , and Y01 are not correlated
            o * 11 11
            » t
            although Xjj and X2 ^ possibly were. Similarly, as can be
            »
            seen in Figure 4, under this transformation and Ha, and
            f
            Y21 are correlated (negatively in this case). Thus, the
            original problem of testing for unequal marginal scales has
            been transformed into the problem of testing for correlation
            I »
            between Y^ and Y2^ • Kepner (1979) suggested the use of
            » T
            Kendall's tau to test for correlation between Y^ and Y9y .
            Kendall's tau was chosen, due to the fact it is a
            U-statistic and, thus, the many established results for
            U-statistics could be applied.
            The test statistic which will be presented in this
            section is very similar to the above mentioned statistic.
            However, when censoring is present, the true observed value
            » » If
            of X ^ ^ O T X2 1
            (or
            both) is not known,
            a nd
            thus
            Y11 Y21
            (or both) are
            also
            affected. To take
            this
            into
            account , a
            modified Kendall's tau will be used which was presented by
            Oakes (1982) to test for independence in the presence of
            censoring. First though, some additional notation must be
            int roduced.
            I I
            Recall, (Xjj,X2i) denotes bivariate random variables
            t 1
            which are distributed as (xii>^21^* Let 0^,02, •••»Cn denote
            the censoring random variables which are independent and
            identically distributed (i.i.d.) with continuous

            21
            Figure 3.
            Contour of an Absolutely Continuous Distribution
            That Has Equal Marginal Locations and Equal
            Marginal Scales under the Transformation.

            22
            Figure 4. Contour of an Absolutely Continuous Distribution
            That Has Equal Marginal Locations and Unequal
            Marginal Scales under the Transformation.

            23
            distribution function G(c) where denotes the value of
            ! I
            the censoring variable associated with pair In
            the case of random right censoring, the observed sample
            f I
            consists of Xx . = min(Xj^,C^) and X2^ = min(X2^ , C^ ) . These
            pairs can be classified into four pair types which are
            Pair Type Description
            f »
            1
            Xli x2i *
            2
            Xii X2i> Ci
            3
            X¡i>Ci>
            X2i 4
            Xii>Ci>
            X2i>Ci
            Consider the following orthogonal transformation applied to
            the observed sample:
            /
            Yli = Xli + X2i an(* Y2i = Xli _ X2i ^or i = 1 ^ , . . . , n.
            Notice that, due to censoring in type 2,3, or 4 pairs, the
            » T
            true values of and Y2^ (denoted Y^ and Y2i , i.e. , the
            values had no censoring occurred) are not actually
            observed. The following table, Table 2.1, summarizes the
            t I
            relationship of the true values of Y ^ and Y2^ to the
            observed values

            Table 2.1
            24
            Summarizing the Relationship Between the True
            t T
            Values of and Y2 ^ to the Observed Values
            Pair
            Type
            Description
            Relationship
            Between
            Yli and Yli
            Relationship
            tBetween
            y21 and Y2i
            1 i X2i Yli
            =
            Yli
            Y2 i
            = Y2 i
            li x2i>Ci
            Yli
            >
            Yli
            Y2 i
            < Y2 i
            li>Ci’
            x2i 1
            Yli
            >
            Yli
            Y2i
            > Y 2 i
            1 i>Ci»
            X2i>Ci
            Yli
            >
            Yli
            uncertain
            or Y21 < Y2i)
            The modified Kendall's tau (denoted CD for
            c^oncordant-discordant) can now be defined as
            CD
            a., b.. where for i < i ,
            ij ij
            r
            r
            1
            if
            Y
            ! Y, .
            1
            if
            Y
            v Y
            l i
            Ij
            2 i
            2j
            â– <
            -1
            if
            Yli '
            Y. . and
            1 J
            bu- •<
            -1
            if
            Y
            2i
            1 Y
            2j
            0
            if
            uncertain of
            0
            if
            unce r t ain of
            V
            the
            relationship
            V
            the
            relationship
            (2.2.1)

            25
            (here Y1i
            I
            can be read as
            i
            Ym is definitely smaller
            than
            For example, if the iL pair is a type 1 and the j
            pair is a type 2 and it was observed that Yli < Ylj. then
            i tit
            aij = 1 since Y^ = Y^ and Y^j < Y^ (thus Y^ < Y^). If
            Yli > Ylj had been observed, then a^^ = 0, since the
            t f
            relationship between Y^ and Y^j is uncertain. Similarly,
            if Y2¿ < Y2j ’ then f'ij = 0» since Y2¿ = Y2i and Y2j < Y2j
            » I
            (thus, the relationship between Y2i and Y2j is uncertain).
            On the other hand, if Y2i > Y2 j had been observed, then
            b^j = -1 (by a similar argument).
            Table 2.2 summarizes the necessary conditons for a^^
            and b^j to take on the values of -1, 1 or 0. The product of
            a^j and b^j results in a value of 1 if the it'1 and j*"*1 pairs
            of the transformed data points are definitely concordant, a
            value of -1 if the pairs are definitely discordant and 0 if
            it is uncertain. If the iC^ pair is a type 4 (i.e., both
            » I
            Xu and ^2^ were censored) then b^j will always be 0 since
            the relationship between the i n and jcn pair is always
            uncertain regardless of the pair's type. Thus, type 4
            pairs always contribute 0's in the sum for CD. Notice, also
            in the case of no censoring this modified Kendall's tau
            reduces to the Kendall's tau applied to the transformed
            data, the statistic investigated by Kepner (1979).

            26
            Table 2.2 Summarizing the Values of a.- and b.. for i = 1: if and one of the following occurs,
            ith pair type
            Jth
            pair type
            1
            1
            1
            2
            1
            3
            1
            4
            a . .
            ij
            = -1 :
            if Y^ > Y^j and one
            of
            t he
            following occurs,
            t" Vi
            i pair type
            3th
            pair type
            1
            1
            2
            1
            3
            1
            4
            1
            a . .
            ij
            = 0:
            for all other cases
            bij
            = 1 :
            if Y2^ < Y2j and one
            of
            the following occurs,
            ith pair type
            3th
            pair type
            1
            1
            2
            1
            1
            3
            2
            3
            bij
            = -1 :
            if Y2^ > Y2j and one
            of
            t he
            following occurs,
            ith pair type
            3 th
            pair type
            1
            1
            1
            2
            3
            1
            3
            2
            bU
            = 0:
            for all other cases

            27
            Next, we establish some properties of the CD statistic.
            Lemma 2.2.1: Under HQ ,
            E ( C D ) = 0
            and
            Var(CD) = a + — —
            n ( n - 1)
            n ( n - 1 )
            where
            a nd
            a = 4P(a . .= 1,b. . = 1 )
            ij iJ
            2p ■bl3‘
            1 , a . . , =
            ij
            f N
            *—H
            II
            •*-)
            *H
            x¡
            + 2p = 1 , a . . ,
            ij
            = -1,b . . , =
            ij
            1)
            + 4?^.=
            = -1,b . . , =
            ij
            1)
            - 2P(a1j =
            ‘•blJ-
            = 1,b. . , =
            ij
            1)
            - 2PU1J*
            = -1 , a . .
            ij
            -i.bir
            = 1)
            - 4P(a1;j-
            ‘•bU-
            1 *au •"
            ’bij *=
            1)
            (2.2.2)
            Proof:
            Throughout this proof, Theorem 1.3.7 in Randles and
            Wolfe (1979) will be used extensively and thus its use will
            not be explicitly indicated.
            Under H„
            ° >
            (Xli,X2i,X1j ,X2j ,Ci ,Cj ) = ^X2i’X1i’X2j ’X1j ’ Ci ’ Cj^

            28
            and therefore it follows that
            (Xli’X2i’Xlj»X2j’0i>6j} = (X2i,Xli,X2j,Xlj,f(6i),f(6j))
            (2.2.3)
            where
            Xu = min(xJi,Ci),
            »
            X2i = rain(X2i ,C±),
            ó ^ indicates what type of pair (X^.X^) is,
            and
            f (6i) indicates what type of pair (X2i,X^) is.
            Thus, f(*) is the function defined below.
            5i
            f (6i)
            1
            3
            2
            4
            Let Yli Xli + X2i a nd Y2i Xli X2i; thus from (2.2.3)
            (Yli>Y2i»Ylj>Y2j>6i>6j) = (Yli,-Y2.,Ylj,-Y2j,f(6i),f(6j))
            Applying the definition of a^ and b ^ j in (2.2.1) (or using
            Table 2.2) to the above, it follows that
            and thus
            p
            29
            and
            p(aij ■ -1> bij ‘ ' Pbij ‘ -1J
            (2.2.4)
            Now,
            E ( C D ) = —— I E ( a . . b . . )
            (?) i where
            Eta^bjj) - (DF^jbjj - 1) + (-l)P(aijblj - -1)
            " p - P(atj - l.bjj - -1) - P(ai:j - -l,btj - 1) .
            Applying (2.2.4) to the above, it follows that E(a^^.b^j) = 0
            and thus E(CD) = 0.
            Note, that under H Q ,
            <'Xli»X2i*Xlj ,X2j »6i><5j) ^Xlj »X2j »xii»x2i»5j > and thus
            (Yu,Y2i,Yij,Y2j.íi,6J) - <í1J.'f2j.íli.'í2i-5j'6i> '
            (2.2.5)
            Applying the definition of and b^j as before, it follows
            (aij >bij } ( aij * bij }
            and also

            30
            and
            P(ai3 - l,b1;¡ - 1) - P(a13 - -l.b^ - -1)
            P * P Now,
            Var(CD) = [——] Var( £ a b..)
            ' ^ 1 i !J 1 J
            (J) ^<3
            , 2
            = [ 1 T y Cov ( a . . b . . , a . , . , b , . , )
            r?i i The three possible cases to consider for the covariance are
            1 ) i^i ' , j ¿j ' , i 2) i = i' , j=j ' , i< j , i ' and
            3) where exactly two of the four subscripts
            i Case 1) i' , j ¿j' :
            In this case, Cov(a^jb^j, a^ij,b^iji) = 0 since the
            bivariate pairs are i.i.d.
            Case 2) i=i', j =j':
            In this case ,
            Cov(aijbij , a^bjj) - EK^by) ]
            ■ P(aij * ‘>bij ' »> + P(aij ‘ 1 ■ bij ' -*>
            + P(a13 - -l.b^ - -1) + P(a13 - -l,b13 - -1)
            = 4 p(aij = l»bij = 1) = 4a (by part a).

            31
            Case 3) Exactly two of the four subscripts i the same.
            Now,
            Cov(atj bij
            ’ aikbik)
            - E(atjb
            i j aik
            the following
            events:
            Al,!:
            í a i j "
            1’ aik =
            1}
            Al,-1:
            {aij =
            aik °
            -1}
            A-1 , 1:
            {aij =
            _1» aik
            = 1}
            A-1,-l
            : {a±J =
            -1’ aik
            = -1}
            and similarly define the events , B_^ ^ and
            B_j Using this notation, E ( a ^ ^ b ^ ^ a ^ ) can be written
            as
            E(aijbijaikbik) =
            l I l I (-l)k+I+"+"p(A
            k = 0 A-0 m = 0 n =0 (-1) ,(-l)x
            , B )
            (-l)m,(-1)n
            (2.2.6)
            Table 2.3 describes the events A , and B
            <-i)k,(-Dt <-i)m,<-nn
            in more detail and the restrictions placed on the 6's.
            Now, to simplify the probabilities in (2.2.6). Note,
            under Hq
            d
            ^X2i,Xli'X2j *Xlj
            X2j ,Xlk’X2k,,Si’5j ’ ^ k ^
            X2k>Xlk>f(6i)>f

            32
            Applying the transformations
            Yli = Xli
            X2i and Y2i Xli X2i
            it f ollows that
            6i.6j'Sk>
            Now, applying the definitions of a^j and in (2.2.1)
            using Table 2.2), notice that if b^^ = 1 (i.e., ^2.£ 6 ± e ( 1 , 2 ) and 6 . e ( 1 , 3 ) , then -Y^ > -Y±. , f ( 6 . ) e ( 1 , 3 )
            f(6j)e(l,2) which would yield b^j = -1.
            Using similar arguments, it follows
            and
            (a..,a.,,b..,b.,) = (a..,a..,-b..,-b.,)
            ij ’ ik’ ij ’ lk ij ’ ik’ ij ’ lk
            and thus
            P(A,
            1 9
            1
            ,B 1,1 ) =
            p(Ai,i
            -1
            P ( A ,
            *â–  9
            1
            ,B-1,1)
            - P (A.
            1 9
            1 '
            B1
            P(A_1
            9 ~~
            1 ,B 1 , 1 )
            = P(A_
            1-1
            9
            P ( A _ x
            9 ~~
            1 *B-1,1
            ) = P (A
            -1 ,
            -1
            ’ B 1 , - 1
            P(A_X
            , 1
            ,B 1 , 1 )
            = P(A_1
            , 1
            P(A_X
            , 1
            ’B-l,1}
            = P (A_
            1 , 1
            )
            P ( A ,
            1 9
            -1
            »B 1 , 1 ^
            = P ( A ,
            1 9
            -1
            > B
            -i.-i’
            P ( A,
            L 9
            -1
            ’B-l,1)
            = P ( A :
            9
            Bi,-i>
            (or

            Table 2.3 Describing the Events A and B and
            Event Description
            A1,1B1,1
            (Yli
            (Y2i A-1, 1B1,1
            (Y11>Y1jâ– Yli
            A1,-1B1,1
            Yik)
            A1, 1B-1,1

            Y2j
            A1,1B1,-1

            A-1,-1B1,1
            Y1j•Yli>Ylk>
            (y21 A-1,1B-1,1
            Ylj'Yli
            Y2J
            A-1,1B1,-1
            (Y1i>Y1J,Yii (y21 A1,-lB-l,1
            (YliYlk>
            Y2j
            A1,-1B1,-l
            Ylk>
            (Y21 A1,1B-1,-l
            ^ Yli Y2j
            A-1,-1B-1,1
            (Yu>YirYii>Yik>
            Y2j
            A-1,-1B1,-l
            Ylj•Yli>Ylk)
            A-1,1B-1,-1
            Ylj-Yli
            (Y2i>l2j
            A1,-1B-1,-l
            Ylk)
            (Y21>Y2j
            >
            1
            1
            cc
            1
            1
            Ylj'YU>Ylk>
            Y2j
            the Restrictions on the
            6 ’ s
            Restrictions
            on the
            6 ' s
            6 j
            Y2i 1
            1,3
            1,3
            Y2i 1
            1
            1,3
            Y2i
            1
            1,3
            1
            Y2i 1
            1,2
            1,3
            Y2i>Y2k>
            1
            1,3
            1,2
            Y2i
            1,2
            1
            1
            Y2i 1
            1
            1,3
            Y2i> Y2 k >
            1
            1
            1,2
            Y2i
            1
            1,2
            1
            Y2i>Y2k>
            1
            1,3
            1
            Y21>Y2k>
            1
            1,2
            1,2
            Y2i 1
            1
            1
            Y21>Y2k)
            1
            1
            1
            Y2i>Y2k>
            1
            1
            1,2
            Y2i>Y2k)
            1
            1,2
            1
            Y2i>Y2k>
            1,3
            1
            1

            34
            Similarly, under HQ
            (Xli,X2i,Xlj,X2j,Xlk,X2k,<5i’5j’lSk^
            = (xii»X2i>xlk>X2k’Xlj’X2j»6i>6k’6j)
            and applying the definition of a^ and b^j in (2.2.1) it
            follows that
            (aij >aik’bij >bik) " (aik’aij )bik’bij}
            This yields that
            P »B-1, i^ = p(Ai,-i
            » B 1 , -1
            hd
            >
            1
            ►—*
            »Bi ,-P = p(Ai ,-i
            » B - 1 , 1
            p ,B i , i ) = P(A_1 ^l
            ’B1 , 1 )
            p(ai,-i
            ’ B -1 ,-P = P(A-1 ,
            1 »B-1,
            Thus, ^^aij^ijaik^ik^ can reduced to a sum of six terms
            instead of the original sixteen; i.e.
            E(ab..ab ) =
            ij ij ik ik
            1111
            l l l l (-D
            k = 0 £ =0 m = 0 n =0
            k+£+m+n
            p (A. . . ,B )
            (-i)k,(-i)* (-n“,(-i)n
            ' 2P(A1,1 â– Bl,l) + 2PU-1,-1 + 4P(A1,-1
            ' 2P " 4
            Note, the subscripts are arbitrary; thus
            E(aijbijaikbik) = E(aijbijakjbkj) = E(aijbijajkbjk)

            35
            and therefore combining the results from case 1, 2 and 3, it
            follows that
            !-l ) Y}
            Var(CD)
            â–¡
            As seen in Lemma 2.2.1, the variance of CD depends on
            the underlying distribution of and possibly C.
            Therefore, CD is not distribution free under HQ. Section
            2.3 will discuss a permutation test based on CD that is
            conditionally distribution free. This test is recommended
            for small samples. For larger samples, Section 2.4 presents
            the asymptotic normal distribution of CD using a consistent
            estimator of the variance. This result can be used to
            /
            construct a distribution free large sample test based on CD.
            2.3 Permutation Test
            In the situation where the sample size is small, a
            permutation test based on CD is recommended. What is
            considered a small sample size will be discussed in Chapter
            Five when the Monte Carlo results are presented. Now, we
            will develop the motivation for the permutation test.
            Recall, under HQ
            f
            (X
            2i,Xli>Ci>

            36
            and thus
            x2i
            ,«i) = (X21,Xli,f(6i))
            (2.3
            where
            Xli =
            »
            min(Xli,Ci), X2i
            = min(X2i,Ci), 6^ is the
            pair
            type
            ( i . e .
            6 ^ = 1,2,3 or 4)
            and f(<5^) is a function
            such
            that
            1
            2
            3
            4
            f (6±)
            1
            3
            2
            4
            0
            Let k = | j be an operator such that
            r
            (Xli,X2.,6i)k = ^
            (X1i’ X2i»6i)
            if k = 1
            (X2i>Xli»f(6i)} if k = 0
            and K = {k: k is a 1 x n vector of 0's and l's} (of which
            there are 2n different elements). Thus, applying this
            operator to (2.3.1), we see under HQ, P{(X^,X2^,6^) =
            (Xu,X2i,6i)0} = P((Xli,X2i,6i) = (X1.,X2i,6.)1}. Applying
            this idea to the entire sample (in which the observations
            are i.i.d), under H , it follows that
            {(Xjj,X2j,6^) 1,(X^2,X22,i52) 2,...,(X^n,X2n,<5n)kn}
            t ! I
            — i(X-j],X?^,6^) 1 » ( Xi 2 , X2 2 , 6 2 ) 2,...,(X,,X2n,6n) n}
            (2.3.2)
            • • •

            37
            where k and k' are arbitrary elements of K. Therefore,
            unde r HQ, given
            { (x
            11» x21’^1^ >(xi2,x22’^2^’* * * »^ x1n’x2 n’ 6n)>,
            the 2n possible vectors
            { (xii»x21»^l) l»(xi2’x2 2’^2^ ^‘’^xln’x2n’^n^ n}
            are equally likely values for
            { ( X j i >X£i , <5 ^ ) ,(Xi2,X22> The idea of the permutation test is to compare the
            observed value of CD, for the sample witnessed to the
            conditional distribution of CD derived from the 2n equally
            likely possible values of CD (not necessarily unique)
            calculated from
            { (xii ,x21 1>(x12’x22’^2^ 2,...,(xin>X2n»(5n^ n ^ •
            Note, since the sample observed is censored, the 2n
            vectors {xii>x21’^l^ l,(x^2»x22’^2^ ^’‘''’^xln’x2n’^n^ n }
            are not necessarily unique. If a pair is a type 4 (i.e.,
            both and X2j were censored), then (xij >x2j >^j ) ^ =
            (xjj,x2j , 6j ® . In fact, there are only 2^n-n4^ unique
            vectors (n^ = number of type 4 pairs), since P(X,^ = ^2^) =
            0 if ^xii»^2i^ is not a cYPe ^ pair under assumption A2. As
            a result, the permutation test, in effect, discards the type
            4 pairs (since a j b ¿ j = 0 if the i c or j C ^ pair is a type
            4) and treats the sample as if it were of size n-n^ with no
            type 4 pairs occurring.
            With regards to the transformed variables (^ii*^2i^
            i = 1,2, ...,n, the permutation test can be viewed in the
            following way. Consider the transformations = X^ + X2^

            38
            and Y2^ = - ^2i* Applying these to (2.3.1) and (2.3.2),
            we see that under Hq
            (Yli,Y2i,6i) = (Yli,-Y2i,f(6i))
            and similarly,
            { ( Y
            11 ’ L21
            Si)
            (Y
            1 2 * 12 2
            s2)
            (Y
            In’ 2 n
            5n>
            n-,
            »
            !
            k
            whe re
            k.
            (Yii,Y2i,5i)
            if k = 1
            (Yii,Y2i,Si) = ^
            (Yli’“Y2i’f(6i}) if
            k = 0
            and k and k' are arbitrary elements of K. That is, under
            H0, given { (y:2 ,y21,Sx),(y12,y22 , s2),..., (yln,y2n,sn)}, the
            2n possible vectors
            kl k2 kn
            ^yl1 ’y21’61) >(yi2’y22’^2^ ’ " *’^yln’y2n’^n^ } are
            equally likely values for
            { ( Y j ^ »y2x > ^ i ^ > ( Y i 2 > Y2 2 ’ ^ 2 ^ ’ * * * ’ ^Yln’Y2n’^n^
            To perform the permutation test, the measurements
            (x1i*x2i’^i) ^ = l»2,...,n are observed and the
            corresponding value of CD is calculated. Under Hq, there
            are 2n equally likely transformed vectors for
            {(Yll’Y21’6l)’(Y12’Y22’62)’“*’(Yln’Y2n’
            39
            statistic is computed for each of these possible vectors and
            from this the relative frequency of each possible CD value
            is determined. The null hypothesis is rejected if the
            original observed CD value is too large or too small when
            compared to the appropriate critical value of this
            conditional distribution.
            2.4 Asymptotic Results
            In Section 2.3, a permutation test was presented to
            test Hq, when the sample size was small. In larger sample
            sizes, the permutation test becomes impractical and time
            consuming. In these situations, the asymptotic results
            which will be presented in this section could be employed.
            Theorem 2.4.1: Under Hq,
            — ——* N(0,1) as n ■* « ?
            [ Var(CD) ]X/2
            where
            Var(CD)
            2 4(n-2)
            n(n-l) a n(n-1) Y
            Proof;
            Note that CD is a U-statistic with symmetric kernel
            h(X^,Xj) = Thus, by applying Theorem 3.3.13 of

            40
            Randles and Wolfe (1979), it follows that
            CD
            + N(0 ,
            as n > oo
            whe re
            ?1 = E[h(X.,Xj)h(Xi,Xk)]
            Note that
            = E[aijbij aikbik]
            il
            n
            2 4(n-2)
            n(n-l) a n(n-l) ^
            Y
            as n + «>,
            therefore after applying Slutsky's Theorem (Theorem 3.2.8,
            Randles and Wolfe, 1979)
            CD
            [Var(CD)
            —+ N(0,1) as n + °°.
            â–¡
            Corollary 2.4.2: If Var(CD) is any consistent estimator of
            Var(CD), then
            —— —— + N(0,1) as n -*■ °°.
            [ Var (CD) ]^2
            Proof;
            This follows directly from Theorem 2.4.1 and Slutsky's
            Theorem. ^
            Next, we consider the problem of finding a consistent
            estimator for Var(CD). There are many consistent estimators

            41
            for a variance, but three which worked well in the Monte
            Carlo study are described in the following lemma.
            Lemma 2.4.3: Under Hq , the following are consistent
            estimators of Var(CD):
            1) Va r x(CD)
            4
            n
            lllA
            1 ijk
            y
            where
            AijkBijk = (aijbijaikbik + aikbikajkbjk + aijbijajkbjk)>
            2) Var2(CD)
            - 4 (
            l l l + l I 1 ' ) >.
            n n(n-l)2 lii and
            3)Var3(CD)
            — í[-¿- l l («i,!»!,)2] - n(n-l) ( 2 ) 1 4(n-2) rn * ,,
            + — {- Var2(CD)}
            n(n-1) 4
            Proof:
            First, it will be shown that nVar^(CD) —4y.
            nVa r ^(CD)
            lllA
            1 < i < j i jkBijk
            }
            Now,

            42
            , A . B .
            - * Hr I l l
            1 which shows that nVar,(CD) = 4U where U is a U-statistic
            J- n n
            of degree 3 with symmetric kernel h* = A^^^3^^^/3 . Thus, it
            follows that nVar^(CD) —— 4y since y by Hoeffding's
            Theorem (Hoeffding, 1961).
            Next, it will be shown that
            n( Var2 (CD) - Var^CCD)) —0 as n °°.
            First though, notice that Var2(CD) is equivalent to
            2)Var1(CD) + -{ [ l l (a b )2] - (CD)2}
            (n-1) n n(n-l) l Thus ,
            (n-2 )
            n(Var2(CD) - Var^CD)) = n{—“ f ' - 1} Var^CD)
            (n- 1 )
            + 4{[
            l l (a b )2] - (CD)2}
            n ( n-1 ) 2 1 < i < j < n 1 j
            (n-2 )
            (n-1)
            - U U* + 4 {[ l—— l l (a b )2] - (CD)2}
            (n-l)(") 1 —0 as n > oo.
            Therefore, Var2(CD) is a consistent estimator for Var(CD).
            Lastly, it will be shown that
            n(Var^(CD) - Var2(CD) ) —* 0 as n + «.

            43
            Now,
            n(Var3(CD) - Var2(CD))
            u4
            i i (n-l) (") 1 + {(n 2) - 1} nVar2(CD)
            (n-l)
            -> 0 as n -*â– 
            â–¡
            Next, we provide a brief explanation of each of these
            estimators. As was shown in the proof of Lemma 2.4.3,
            ^ 4 * *
            Var,(CD) = — U where U is a U-statistic which estimates y.
            i n n n
            Thus, Var^(CD) is estimating the asymptotic variance of
            CD. Var2(CD) is also estimating the asymptotic variance of
            CD, but in a slightly different manner. Recall, from basic
            U-statistic theory that y is the variance of a conditional
            expectation (Randles and Wolfe, 1979, p. 79) (i.e.,
            y = Var[(ajb^)*] where (ajbj)* = E[a 1 2b^2 ¡ (Y11,Y21)]) .
            Thus, in Var2(CD), for each (Y^,Y2^), the conditional
            expectation is estimated using all the other (Y^,Y2j)'s,
            j*i and then the variance of all these quantities is
            calculated. That is,
            where
            Var-(CD) = - ] {(a.b.)* - CD}2
            4 n t l l
            n-l j*i J
            In contrast to Var^(CD) and Var2(CD), Var3(CD) is
            estimating the exact variance of CD (2.2.2) derived in

            44
            Section 2.2. It is using an estimator of y from Va^CCD)
            and estimating a with a difference of two U-statistics which
            is estimating
            Again, although under HQ , ECa^b^j) = 0, the sample
            estimate for Eia^b^j) (i*e., CD) was left in to possibly
            increase the power of the test under the alternative.
            Each of these variance estimators will be considered in
            the Monte Carlo study in Chapter Five. Although the
            calculations look overwhelming if performed by hand, they
            are all easily programmed on the computer. (See the CDSTAT
            subroutine in the Monte Carlo program listed in Appendix 2.)
            2.5 Comment s
            This chapter has presented a statistic to test the null
            hypothesis of bivariate symmetry versus the alternative that
            the marginal distributions differ in their scale
            parameters. For small samples, a permutation test is
            recommended. A basic disadvantage of this is that it
            generally requires the use of a computer for moderate sizes
            (otherwise it is very time consuming to derive the null
            distribution). For larger sample sizes, it is recommended
            C D
            that be used as an approximation for
            [Var(CD)f^2
            CD
            Thus, for an a level test using the

            45
            asymptotic
            rej ected if
            distribution,
            i CD
            ^ Vo
            [Var(CD)] 2
            the null hypothesis would be
            > Z
            a/2
            where Z is the value in
            a / 2
            a standard normal distribution such that the area to the
            right of the value is a/2.
            Chapter Five will present a Monte Carlo study which
            uses the asymptotic normal distribution of CD (with a
            consistent variance estimator) to investigate how well the
            test performs under the null and alternative hypotheses.
            First though, some comments on this chapter.
            Comment 1
            One possible advantage of the CD statistic is the fact
            it utilizes information between censored and uncensored
            pairs whenever possible. In the permutation test, type 4
            pairs have no effect on the outcome of the test. That is,
            they can be ignored, treating the sample as if it were of
            size ni+n2+n3* This is understandable since Xli = X2i = Ci
            and thus they supply no information about the scale of
            relative to
            In the asymptotic test, if one estimated the variance
            in (2.2.2) by estimating a and y with their sample
            quantities (for example, a =
            n(n-l) 1 < i < j < n
            l l (aijbij) ic

            46
            is easily shown that the type 4 pairs have no effect on the
            value of the test statistic. That is, the value of the test
            statistic remains the same whether the type 4 pairs are
            discarded or not. If a different estimate for the variance
            is used, there is a slight change in the test statistic's
            value if type 4 pairs are discarded, due to the different
            variance estimator. Asymptotically, this difference goes to
            zero, due to the fact the variance estimates are all
            estimating the same quantity. Thus, in some sense, the
            asymptotic test behaves similarly to the permutation test
            with regards to type 4 pairs.
            If a and y are known, they are a function of whether
            type 4 pairs are included or not. That is, if type 4 pairs
            were not included in calculating the test statistic (thus
            n=n^+n2+n2), the value for a and y would be larger than the
            value had type 4 pairs been included (since type 4 pairs
            only contribute 0's and never l's or -l's). The effect of
            type 4 pairs on a and y is such that the test statistic's
            value would be the same (or at least asymptotically the
            same) whether type 4 pairs were discarded or not.
            Comment 2
            A disadvantage of the test is that for small samples CD
            is not distribution free. Thus, the permutation test,
            conditioning on the observed sample pairings, must be

            47
            performed to achieve a legitimate distribution free crlevel
            test .
            Comment 3
            It is unclear how the CD statistic would be affected if
            the marginal distributions of and X21 have different
            locations. It is possible that the assumptions made on the
            censoring distribution might not be valid (in particular
            assumption A4, which assumed the same censoring cutoff for
            X^ and ^2i^ or even if this is true, that CD does not
            perform well in these instances. Chapter 5 will investigate
            this problem in further detail

            CHAPTER THREE
            A CLASS OF TESTS FOR TESTING FOR DIFFERENCES IN SCALE
            3.1 Introduction
            In the previous chapter, a test statistic was presented
            to test the null hypothesis of bivariate symmetry against
            the alternative that the marginal distributions differ only
            in their scale parameters. A shortcoming of the statistic
            was the fact the variance of CD depended on the underlying
            distribution and, thus, for a small samples a permutation
            test had to be done or for large samples the variance had to
            /
            be estimated. In this chapter, two test statistics will be
            presented which are nonparametrica1ly distribution-free
            (conditional on N^ = n^ and Nc = n2+n^) for all sample sizes
            to test the null hypothesis of bivariate symmetry. The
            alternative hypotheses are structured by assuming the
            samples come from a bivariate distribution with c.d.f.
            *1 - U x2 - p
            F( , ) where F(u,v) = F(v,u) for every (u,v)
            a 1 °2
            2
            in R . Tests are developed for both of the following
            alternatives to the null hypothesis of bivariate symmetry:
            48

            49
            Case 1 . = ^2 known,
            Ho: ai = a2 an<^ Ka: o< a 2
            That is, the marginal distributions have the same known
            f
            location parameter but, under Hfl, X2 ^ has a larger scale
            I
            parameter than X^ . A possible contour of an absolutely
            continuous distribution of this form was given in Figure 2.
            Case 2 . p j = pi 2 unknown,
            a 1 = a 2 and Ha
            a L < a 2
            Here, the marginal distributions have the same unknown
            »
            location parameter but, under Ha, X2 ^ has a larger scale
            parameter than X^
            (Note, for both cases, the alternative has been stated in
            the form for a one sided test. The procedure which will be
            presented can easily be adapted for the other one-sided or a
            two sided alternative. The latter is discussed at the end
            of this chapter . )
            In Sections 3.2 and 3.3, tests statistics for Case 1
            and Case 2, respectively, will be presented which are
            nonparametrically distribution-free conditional on N ^ = n ^
            and Nc = n,+n^. In both cases, the test statistics can be
            viewed as a linear combination of two independent test
            statistics T and Tn , where T is a statistic based only
            nl nc nl
            on the n^ uncensored observations, while Tn will be a
            statistic based on the n£ = n2 + n3 type 2 and 3 censored
            observations. The conditioning of the random variables

            50
            and Nc on n^ and n2+n^ (respectively) is used throughout
            Section 3.2 and 3.3 and, thus, this condition will not
            always be stated but will be assumed with the use of n^, n2
            and n7. Thus, the test statistics will be written as T
            3 ’ ni , n(
            and TMn n (for Section 3.2 and 3.3, respectively) which
            1 ’ c
            imply conditioning on = n^ and Nc = nc = t^ + n^. Section
            3.5 will consider the asymptotic distribution of each test
            statistic .
            3.2 p ^ = p 2 > Known
            This section will begin by introducing the notation
            necessary for the statistic Tn n designed for the
            1 * c
            alternative in case 1. Recall, the sample consists of
            (Xli,X2i) i = l>2,...,n where = min(X^,C^) and
            I
            X2i = m;>-n (^2 i ’ ^ i ^ * These pairs were classified into four
            types .
            They
            were the
            following
            •
            Pair
            Type
            Description
            Number of Pairs
            in the Sample
            1
            X1i X2i nl
            2
            Xli X2i>Ci
            n2
            3
            Xii>Ci’
            X2i n3
            4
            xU>ci.
            X2i>Ci
            n4
            — n^ + 02 +
            where n
            n3 + n4

            51
            For convenience and without loss of generality, let the
            type 1 pairs occupy positions 1 to n^ in the sample (i.e.,
            {(XjJ,X21),(Xj2,X22(Xln ,X2ni)} ) in random order.
            Similarly, the type 2 and type 3 pairs will be assumed to
            occupy positions n^+1,n^+2,...,n^+nc in random order.
            Lastly, the type 4 pairs occupy positions
            n^+nc+l,n^+nc+2,...,n. What is meant by random order, is
            that the exchangeability property still holds within the n^
            type 1 pairs, within the n2 + n2 type 2 or 3 pairs and within
            the n^ type 4 pairs. This could be accomplished, if the
            pairs were placed into their respective grouping (type 1, 2
            or 3, or 4) arbitrarily, with no regard to their original
            position in the sample. Much easier, from a researchers
            point of view, would be to place the pairs into their
            respective groupings in the same order they occurred in the
            sample (i.e., the first uncensored pair is placed into the
            first position among the n^ uncensored pairs, the second
            uncensored pair into the second position, etc.) . This
            procedure would not affect the desired exchangeability
            property, as deduced from the following argument. In using
            the second method, the reseacher is actually fixing the
            position of the type 1 pairs, type 4 pairs and type 2 or 3
            pairs. Thus, instead of n! equally likely arrangements of
            the original sample, there are n^!n^!(n9+n2)! equally likely
            arrangements when the positions and numbers of the pair
            types are fixed. Therefore, it follows, that each of the
            n^! arrangements of the n^ uncensored pairs is equally

            52
            likely and that the exchangeability property still holds
            within the type 1 uncensored pairs. Similar argueraents for
            the ^2 + 1*2) type 2 or 3 pairs and the n^ type 4 pairs
            hold .
            The following notation will be used in the statistic
            Tn^, a statistic which is based on the n^ type 1 pairs.
            Define a variable to be
            Zi " I X 2 i y I “ I X1 i 11 I for i_1’2’...»n1
            where p is the known and common location parameter. Let
            be the absolute rank of for i = 1 , 2 , . . . , n ^ , that is, the
            rank of |Z ^| among {|z^|,|Z2|,...,|zn j} and let be
            defined as
            H'i = 'R C Z ±) = «
            1 if Z± > 0
            0 if zi < 0
            Note, the variable Z ^ is defined only for the uncensored
            pairs
            The statistic T is then
            nl
            ll
            R
            : = l v. „
            ni i¿i 1 1
            the Wilcoxon signed rank statistic computed on the Z^'s.
            Notation will now be introduced for the statistic Tn ,
            c
            a statistic based only on the type 2 and type 3 censored
            pairs. (The pairs in which only one member has been
            censored.) Define Qj to be the rank of Cj among
            ^ Cn1 + 1*Cn1+2’ ,Cni + nc* and

            53
            YJ‘ i
            ü h
            1 if the j pair is a type 2 pair
            0 if the j11*1 pair is a type 3 pair
            for j = n^+1,n^+2,...,n,+n^. The statistic Tn is defined
            â– 1 c
            a s
            n, + n
            1 c
            C = l y. Q.
            nc j=n1+l J J
            = y ranks of the C's for the type 2 pairs
            A brief explanation of the logic behind the test
            statistic will be presented. For the test statistic Tn , if
            X2 has a larger scale parameter than Xj (i.e., under ) ,
            then 1^2^ - p| - |x^ - p| should be positive and large.
            Thus, the test statistic Tn would be large. In contrast,
            if X2 and X: have the same scale parameter (i.e., under HQ ) ,
            then I X 2 ^ ~ u| ~ j X1i - y| would be positive approximately
            as many times as negative with no pattern present in the
            magnitudes of jx2¿ - p| - |X^ - p|. Thus the test
            statistic would be comparatively less.
            For the test statistic T , if H is true, there should
            nc a
            be a preponderance of type 2 censored pairs (relative to the
            number of type 3 censored pairs) and these pairs should have
            the more extreme censoring values. Figure 5 illustrates
            this idea. Thus, the test statistic Tn would be large. In
            contrast, if H is true, the number of type 2 pairs should

            54
            Figure 5 .
            Contour of an Absolutely Continuous Distribution
            That Has Equal Marginal Locations and Unequal
            Marginal Scales with Censoring Present.

            55
            not dominate n and the test statistic T should not be
            c nc
            unusually large.
            Now we establish certain distributional properties for
            T_ and T_ .
            n. n„
            Lemma 3.2.1:
            distribution
            Conditional on n^ , T has
            as the Wilcoxon signed rank
            the same null
            statistic.
            Proof:
            First it will be shown that conditioning on the n^ type
            1 pairs does not affect the exchangeability property (i.e.,
            (Xli’X2i’Ci) “ (X2i’Xji»Ci)) still holds. Let
            W±= (x|i,X2i,C1) and W* = (x'2i ,Xl± ,C±) and
            G (t) =P(xJt< t1>X2i< t2’Ci< t3) = < t)] where
            I(W < t) =
            if xu< t1,x2i< t2,cjL< t3
            otherwise
            Now, under H, for the entire sample, we have
            f f
            (XU-X2i-Cl> ' (X2i’Xll’Cl)
            and applying an apropriate function (and Theorem 1.3.7 of
            Randles and Wolfe, 1979) thus
            I(W. < t)I(5.= 1) = I(W* < t)I(f(6.) = 1) .
            Taking expectations, it follows that
            E[I(W.< t)I(6.= 1)] = El I(W * < t)I(f(5.) = 1)] .
            Now, recalling that 5^= 1 iff f(5^) = 1; thus
            E[I(5i= 1)] = E[ I(f(6± ) = 1)] ,
            and it follows that

            56
            E [ I (W±< t)I(6jL= 1)] E[I(W% t)I(f(6i) = 1)]
            ' — ——■ ■ •
            E[I(5.= 1)] E[I(f(6±) = 1)]
            This shows that the c.d.f. of given it is a type 1 pair
            is equal to the c.d.f. of Wi given it is a type 1 pair and
            thus the exchangeability property holds within the type 1
            pairs.
            Now, by defining a function
            fj(a,b,c) = |min(b,c) - y| - |min(a,c) - y| and applying
            Theorem 1.3.7 (Randles and Wolfe, 1979, page 16) it follows
            Z - | X 21 ~ y |
            - 1X11 - »l
            = |min(X21>C) - yj
            j »
            - min ( X , C )
            d | ,
            i »
            - |min(X2^ ,C)
            = |min(X1j , C) ~ y 1
            = 1 X1 l - n| -
            1X21 ~ =
            and thus by Theorem 1.3.2 (Randles and Wolfe, 1979, page
            14), the random variable Z has a distribution that is
            symmetric about 0. The proof of Lemma 3.2.1 follows
            directly from Theorem 2.4.6 (Randles and Wolfe, 1979, page
            50) . Q
            Lemma 3.2.2: Under HQ, the following results hold.
            a) Conditional on the fact the pair is type 2 or 3, the
            random variables Yj and Cj are independent.
            b) Conditonal on nc, Tn has the same null distribution
            as the Wilcoxon signed rank statistic.

            57
            Proof :
            First, it will be shown that conditioning on the n c
            type 2 and 3 pairs does not affect the exchangeability
            property. Define W^, ’GW^ £ ^ and I(W^ < t) as in Lemma
            3.3.1. Now under HQ , for the entire sample, we have
            (Xii,X2i,Ci) = (X2^,X^^,C^)
            and applying an appropriate function (and Theorem 1.3.7 of
            Randles and Wolfe, 1979)
            d *
            I(Wi < t)I(5ie(2,3)) = I(W. < t)I(f(5i)e(2,3)).
            Taking expectations, it follows that
            E{l(Wi Recalling that, 6^e(2,3) iff f(6^)e(2,3), and thus
            E[I(6±e(2,3))J = E[I(f(6i)e(2,3))].
            It follows that
            E[I(W.< t)I(6±e(2,3))J E[I(W*< t)I(f(6 . ) e(2,3 ) ) ]
            E[I(5ie(2,3))] E[I ( f ( 5 . ) e ( 2 , 3 ) ) ]
            Therefore, conditional on the pair being a type 2 or 3, the
            exchangeability property still holds.
            Thus, it follows that
            P(Yj = l.Cj < c) = P(xJj < Cj.X^j > Cj.Cj < c)
            = P(X2j < Cj’^lj ^ ^ c) = P ( Y j = 0>Cj * c)
            Noting that,
            P(Yj = l.Cj < c) + P(Yj = O.Cj < c) = P(Cj < c)

            58
            and thus
            2P ( yj = l.Cj < c) = P(Cj < c)
            or that
            P ( Y j = l.Cj < c) = V2 P (Cj < c) = P ( Y j = 1 ) P (Cj < c)
            and thus we see that Yj and Cj are independent.
            To prove part b), let y = (Yn +1 >Yn +2 » • ••»Yn +n
            1 1 1 c
            and
            Q = (Qn ,0n +2 ,...,Qn +n ). By Theorem 2.3.3 (Randles
            11 1 c
            and Wolfe, 1979, page 37), Q is uniformly distributed over
            Rn where
            c
            Rn = {q : q is a permutation of the integers 1,2, ....n^ }.
            Now, let q be any arbitrary element of Rn and let g be any
            “ c
            arbitrary nc vector of 0's and l's. Thus,
            P (Y = g >0 = 5) = P(X = g ) P (Q = <{ ) (by part a)
            and
            P(Y = g )P(Q = g ) = —x
            - C n !
            2 c
            which proves part b). £3
            By Lemmas 3.2.1 and 3.2.2, TR and Tn are
            “1 11 c
            nonparametrica1ly distribution-free conditional on n^ and
            nc, respectively.

            59
            Lemma 3 «2 .3 : Under HQ, the following results hold,
            a) Conditional on n^ , E(Tn^) = n^(n^+l)/4 and
            Var(Tn^) = n^(n^ + l)(2n^ + l)/2 4
            b) Conditional on n , E(T ) = n (n + l)/4 and
            c n c c c
            Var(Tn ) = nc(nc+l)(2nc+l)/24.
            c)
            Conditional on n, and n .
            1 c ’
            independent.
            T_ and T are
            nl nc
            Proof:
            The proof of parts a) and b) follow directly from
            Lemmas 3.2.1 and 3.2.2 and the fact that the Wilcoxon signed
            rank statistic based on a sample of size n has a mean of
            n(n+l)/4 and variance of n(n+1)(2n+l)/24.
            The proof of part c) is also trivial following from the
            fact Tn and Tn are based on sets of mutually independent
            observations. ^
            With these preliminary results out of the way, the test
            statistic T can now be defined by
            1 * c
            T
            nl’nc
            L1 T
            1 nl
            + L2Tn
            nl
            n l + nc
            y 'v. r .
            i=i 1 1
            + L2
            +
            i—i
            II
            •»“)
            where Lj and L2 are finite constants.

            60
            Theorem 3.2.4: Under HQ ,
            »> E - L1E + E2E(Tn >
            i c i c
            = (LjiijCnj + l) + L2nc(nc + 1 ) )/4
            b) Var(Tn n ) = (L^(nj + 1)(2ni+l)
            1 ’ c
            + L|nc(nc+l)(2nc+l))/24
            c) Tn n is symmetrically distributed about E(Tn n )
            and
            d) for fixed constants Lj and L2 , Tn n is
            nonparametrically distribution-free .
            Proof :
            The proof of parts a) and b) follow directly from
            Lemmas 3.2.2 and 3.2.3. To prove part c), it is known that
            the Wilcoxon signed rank statistic is symmetric about its
            mean. Thus, Tn and Tn are symmetric about E(T ) and
            1 c 111
            E(Tn ), respectively. Since Tn and Tn are independent
            c 1 c
            (conditional on Nj = n^ and Nc = nc), the symmetry of Tn n
            follows.
            To prove part d), note that
            P (T
            nl ’ nc
            = k) = P(L1Tn + L9 Tn = k) =
            2 n,
            l P(LlT„1- | L2T - kc)P(L2Tn . k ) ,
            {kc> C
            where {kc}
            l P(LlTn, ' k-kc>p {kc} 1 c
            set of all possible values of E2Tn .

            61
            Now using the nonparametrica1ly distribution-free property
            of Tn and Tn established in Lemmas 3.2.1 and 3.2.2, it
            follows that for fixed L, and L0, L,T„ and L0T are also
            1 Z ’ in, Z n„
            c
            â–¡
            nonparametrically distribution-free
            can be obtained using the fact it is a convolution of two
            Wilcoxon signed rank test statistics' null distributions.
            Thus, for fixed and L2 , the distribution can be tabled.
            Tables in the Appendix 1 give the critical values for Tn n
            with Lj = 1 and L2 = 1 for n^ = 1,2,.. .,15 and
            n£ = 1,2,...,10 at the .01 , .025 , .05 and .10 levels of
            significance. The actual a-levels are also reported for the
            cut-offs given. The decision rule for the test is to reject
            Hq if the calculated test statistic is greater than or equal
            to the critical value given in the table at the desired
            level of significance. A two tailed test (i.e., for H :
            a
            Oj * a2) could be performed by using the symmetrical
            property of the null hypothesis distribution and the table
            to determine the lower critical value for the test
            statistic.
            A test of Hq for larger n^ and n£ can be based on the
            asymptotic distribution of Tn
            which will be presented in
            Section 3.4

            62
            3.3 y^ = y 2> Unknown
            In the previous section, the common location parameter
            was assumed to be known. Generally, this is not the case.
            More often we may assume a common location parameter, but
            this parameter is unknown. This section will present a
            slight modification to the test statistic T_ n to be used
            nl ’ nc
            in these settings. The modification will be to estimate the
            common location parameter using a "smoothed" median
            estimator based on the product-limit (Kaplan Meier) estimate
            of the survival distribution (Kaplan and Meier, 1958). This
            estimated location parameter M, replaces y in the previous
            definitions. That is, define the variable to be
            i = l ,2
            M
            M
            X
            1 i
            The definitons of T . , R, , y . , Q. , T , T and T
            i iii n. n n , , n
            J J 1 c 1 c
            remain unchanged. In this section, the statistic will be
            denoted by TMn n to identify the fact the location
            parameter was estimated with a "smoothed" median estimator
            based on the product-limit estimate of the survival
            distribution. This estimation does not affect the results
            in Section 3.2, but Lemmas 3.2.1 and 3.2.3 c) must be
            reproved, since in the proof of 3.2.1, we utilized the
            independence of the Z^'s, a condition which no longer
            n
            c
            were based on sets of

            63
            mutually independent observations. This is not the case in
            the current context.
            First, we introduce the "smoothed" median estimator and
            the product-limit estimate of the survival distribution.
            Let (Y(l)’Y(2)’-**’Y(2n1+n2+n3)) represent the ordered
            uncensored observations. (This ignores the fact the
            original observations were bivariate pairs, and considers
            only the 2n^+n2+n3 uncensored observations, i.e., 2n^
            components belonging to type 1 pairs, the n2 uncensored
            components of type 2 pairs and the n^ uncensored components
            of type 3 pairs.) That is, X^j = if is uncensored
            and X^j has rank k when ranked among the set of all
            uncensored observations from either (both) components of the
            pairs for i=l,2 and j=l,2,...,n. Let n(^)>
            i = l , 2,...,2n^+n2 + n3, be the number of censored and
            /
            uncensored observations which are greater than or equal to
            Y(i). Thus,
            n
            U)
            2
            = I
            n
            l
            I(X
            i = lj=l
            ij
            Y(i)}
            whe r e
            I is the indicator function which takes on a value of one
            when the argument is true and zero otherwise.
            The product-limit estimate of the survival distribution
            is defined as

            64
            r
            s(t) =
            i
            j
            n (n
            k = l
            if t < Y
            (1)
            no’ 1)/n(k) lf Y(j) < c < Y(j+n for
            j=l, 2 ,...,2n1+n2+n3-l
            if t > Y
            ( 2 n ^ +n 2 -Hn^ )
            (Note, that Y ^ ^ is the smallest uncensored observation and
            Y( 2nj +n2 + n^ ) t^ie larSest uncensored observation.)
            The definition given here assumes no ties in the uncensored
            observations. This is valid under assumptions A2 and A3.
            Using the above definition, the "smoothed" median estimator
            M is
            M =
            where
            and
            ^ S(m L) - 0.5
            + T x (m - m )
            S(mx) - S(m2) Z 1
            <
            if m^ t m2
            if ml = m2 ,
            m1 = mi n { Y^ ± j : S(Y^±jj > V2 }
            m2 = max{Y^. j: S(Yq^ < V2 }
            A brief explanation of this estimator follows.
            The product-limit estimate of the survival function,
            S(t), is a right continuous step function which has jumps at

            65
            the uncensored observations. An intuitive estimate for the
            common median is the value of Y ^ ^ such that S(Y^j) = V-?,
            which often does not exist due to the nature of S(t). Thus,
            the "smoothed" estimator was suggested by Miller (1981, pg .
            75), which can be viewed as a linear interpolation between
            m^ and . If the Y ^ ^ exists, such that S(Y^)) = V2 . then
            m^ = m2 and M is that value of Y^.^ by definition.
            Lemma 3.3.1: The statistic M is a symmetric function of the
            sample observations.
            Proof :
            "fa "fa
            Let (Y^j,Y^2)»•••»Y^2n)^ represent the ordered 2n
            observations where Y^^ < ^(2) * ••• ** ^(2n)’ This again is
            ignoring the fact that the original observations consisted
            of n bivariate pairs and treats the sample as if it
            consisted of 2n observations (some of which are censored).
            Under assumption A2, there are no ties among the uncensored
            observations. Similarly, by assumptions A2 and A3, there
            are no ties between an uncensored and a censored
            observation, although there may be ties (of size two) among
            the censored observations because type 4 pairs contribute
            two components with the same value. The product-limit
            estimator S(t) can be viewed as a function of the vectors
            ^|p ^|p ^|p
            (Y(1),Y(2)* * * *>Y(2n)) and (r(1)»I(2)’ * * • ’l(2n)) where

            66
            (j)
            1 if Y,.. is censored
            (j )
            0 otherwise
            in the fact that
            2 n
            /, \ = y y i(x..> y,, )
            (1) lilj-l ^
            X
            = 2n + 1 - (rank of Y^^ in (Y ( i ) > ^ (2 ) > • ' ' >Y dn)'* ’
            In addition, S(t) can be expressed as
            S(t) =
            1
            0
            t < min Y , . , : I, . =
            1 (i) (i)
            1}
            t > ma x Y , . x : I/.N
            1 (i) (i )
            = 1
            n
            2n - j
            X(j)
            Y* V (j)
            otherwise
            Thus, S(t) is a symmetric function with respect to the
            sample observations and therefore M, being a function of
            S(t), is also. Cl
            Lemma 3.3
            . 2 : Conditional on n^ , Tn has the same null
            distribution as the Wilcoxon signed rank statistic
            Proof :
            Let V = {4^, f2, ...,4'n }, where 4^ = H' (Z ±) and
            R = R, , R „ R I with R. =
            1 1 2 n ^ ‘ i
            Let V be any arbitrary element of
            P = i V : 4' is a 1 x n, vecto
            1 -o -o 1
            absolute rank of Z ^

            67
            (of which there are 2 different elements), and let r be
            any arbitrary element of
            R = {r : r is a permutation of the integers l,2,...,n^}.
            Now, under the null hypothesis,
            (X
            li’A2i
            ,C.) 2 (x
            f
            2 i
            1 i
            i’Ci>
            t f
            and thus letting = min(X^,C^) and X2^ = mi n ( X2 ^ , C ^ ) , it
            follows that
            (X21'XU)
            for i=1,2,.
            Now, let k
            n^ and these pairs are also exchangeable,
            be an operator such that
            (X
            li’
            X2 i )

            if
            k = 1
            (X2f Xll>
            if
            O
            II
            Thus, under Hq and using the exchangeability property, it
            f o 1lows
            l(XU- x21>’ (X12’ X22>>-"'> X2„,>>
            d k1 k2 k
            = {(xlr , x2r ) , (xlr , x2r ) ,...,(xlr , x2r ) ni} .
            1 1
            (3.3.1)
            Recalling that M = the estimate of the location parameter,
            is a symmetric function of the components of the observation

            68
            pairs from Lemma 3.3.1 and defining a function
            fl(yl»y2) = lyl " Ml " ly2 " Ml = Z
            it follows from applying this function to (3.3.1) that
            { Z, , Z , , Z | — ÍZ y Z y . . . y Z
            1 1 2’ n J 1 r ’ r„ r 1
            1 12 n^
            k k k
            = {(zr ) (zr ) ,..., (zr ) ni}
            (3.3.2)
            where (Z ) =
            r .
            i
            if k = 1
            -Z if k = 0
            r .
            Now defining a function Í2(Z) = (Y, R), where Y and R are
            1 x n^ vectors such that
            and
            V
            II
            •»—>
            1
            if Z.
            J
            V
            o
            0
            h
            if Z .
            J
            < 0
            = absolute rank of Z^ ,
            i . e . ,
            rank of j Zj | among
            { | ^ i | >
            1 Z 2 1
            > • • • j j
            Zn,l>
            for j=1,2,...,n,. Applying this function to (3.3.2) it
            follows that
            ( ^ 1 » ^o » • • • > ^_ » R. > R, , • • • , k )
            12 n ^ 1 2 n ^
            = (Y , V , V , R , R , . . . , R )
            r, r ’ r r ’ r ’ r
            12 n^ 1 2 n^
            H kl k? k
            = {('fj. ) , (V r ) >...,('Fr ) "i, R r , R r ,...,Rr }

            69
            where (f ) 1 =
            i
            if k.= 1
            i
            1 - r. i
            i
            Now since k and r were arbitrary vectors, it follows that
            P(Y = T*, R = R*) = P(T = X , R = r) =
            1 1
            x
            1
            1
            Thus noting this produces the same null distribution for the
            Wilcoxon signed rank statistic, the proof is complete. Q
            Lemma 3.3.3: Conditional on n^ and n , Tn and Tn are
            1 c
            independent.
            Proof:
            This proof is done in a series of steps which are
            stated as Claim 1 to Claim 7 in an attempt to avoid
            confusion.
            Let y £ be defined as before and let (x^,c^) denote the
            . v
            observed value of the iz type 2 or 3 pair
            i=n^+l,...,nj+n . Note, one component was censored, and
            thus its observed value was c^ while the other component was
            uncensored and its value is denoted by x^. This is not
            specifying which component (x^ or X2i^ was censored.

            70
            Claim 1: y^ is independent of (x^,c^).
            This follows by noting that under Hq and using the
            exchangeability property of type 2 and 3 pairs (as was
            shown in Lemma 3.2.2) that
            P{yi = l | (xi , ci)} = p{Xli=xi,X2i = ci| (x±,ci)}
            = P{ X1 i = ci , X21 = xi | (xi , ci ) } = P{ Yi=0 | (x^^ , c± )}
            Since P{ Yi = l | , ci ) } + P { = 0 | ( x i , c¿ ) } = 1, Claim 1
            follows. Now define y = (Yn + i, Yn +2»**‘>Yn +n )•
            1 1 1 c
            Claim 2: y is a vector of n i.i.d. Bernoulli random
            variables which are independent of
            ^xnL + l ’ cn1 + l ^ ’ ^xnL + 2 ’ cn1 + 2 ^ ’ * * * ’ (xn1 + nc ‘ cni+nc^
            This follows from Claim 1 and the fact that
            { (xnj + l ’ cnx + l ^ » ^xn:+2 » cni+2 ^ » * * ' » ('Xn1+nc ’ Cn1 + n(;
            )}
            are i.i.d.
            Claim 3: y is independent of
            * ^nj + l » cni + l ^ » ^xnx+2 ’ cn1+2 ^ ’ * * * » (xnL+nc » Cn1+n(.) * ’
            x„ and x„ where
            ~nl ~n4
            ~nl " {UH,X12)’(x12’x22)>---. and
            -n< = {(cni+n +1 ' cn,+n +1)» * * ' ’(cn»cn^}
            4 I C 1C
            (i.e., the observed totally uncensored type 1 pairs and

            71
            the observed totally censored type 4 pairs,
            respectively ) .
            This follows from Claim 2 and the fact y is a function of
            the type 2 and 3 pairs only.
            Claim 4: y is independent of xn
            in,
            5 where x
            ( nc )
            1 ~ “4 - ' "c' "*c
            denotes the observed ordered uncensored
            members of type 2 and 3 pairs and c✓n \ denotes
            ’ nc
            the observed ordered censored members of type 2 and 3
            pairs.
            Note, this claim follows directly from Claim 3 and the fact
            that X/n \ and c/n \ are functions of
            c' ~ ' c '
            ^Xnx + 1 ’ cni + l ^ ’ ^0^2 ’ cnL + 2 ^ ‘ » (xn1+n£ > cn1 + nc^ only .
            Claim 5:
            yc is independent of x , xQ^, x^)
            Y
            and c
            (nc>
            where y„ = {y„ , y
            ~c c(l) c(2)
            element of c^n ^ and yc
            th
            (i)
            to the pair of which c
            (i)
            ”C( c(i) ls the r
            is the y which corresponds
            was a member.
            This claim follows from Theorem 1.3.5 of Randles and Wolfe
            (1979) and since yc is a fixed permutation of y. Note that
            the i.i.d. property still holds for the y 's.
            (i)

            72
            Claim 6: Given xn , xR^ , x(n )
            a random variable; that is
            observed .
            and S(nc)
            Tn is no longer
            the value of T.
            is
            This follows directly from the definition of Tn .
            Claim 7: Note, that
            n , + n n, + n
            r C v C d +
            Tn = l YiQi " l jYc " W
            c j=n1+l J J 3=0^1 (j)
            which shows that T is a function of y and is
            nc 1 c
            independent of x„ , x„ , x, \ and C/„ \.
            -n1 » -n^ * ~(n ^) ~(nc)
            Thus, Tn has a null distribution equivalent to the Wilcoxon
            c
            signed rank null distribution and is independent of T
            nl
            which is a function of xn » xn > x(n ) and C(n \ only. Q
            â– '*1 ~ 4 ~v c ^ ~ v c '
            With the proof of Lemma 3.3.3, Theorem 3.2.4 is valid
            for
            the modified test
            statistic TM„ _ .
            nl ’nc
            That
            is, under HQ
            and
            conditional on n^
            and n , TM„ has
            the
            same
            c ’ nl > nc
            distributional properties stated in Theorem 3.2.4 for T
            nl’nc
            and the tables in the appendix are valid.

            73
            3.4 Asymptotic Properties
            In this section, the asymptotic distribution of the
            test statistic T
            _ (and TM„
            n^ , nr ni ,n,
            ) under H will be
            -c Jl»uc
            established. The asymptotic normality of the test statistic
            will be presented first, conditional on = n^ and Nc = nc
            both tending to infinity and second, conditional on n
            tending to infinity. In the second case, this is the
            unconditional asymptotic distribution since it only requires
            that the sample size go to infinity. Note that, under
            assumption A.5 (A.5 stated that the probability of a type 4
            pair is less than one), as n °°, N^ + Nc = (n - number of
            type 4 pairs) -*â–  <*> also. The asymptotics will be presented
            for the test Tn n only. In the previous section, it was
            shown that under H and conditional on N, = n, and N„ = n
            o lice
            T and TM have the same null distribution; that is
            nl’nc , nl’nc
            d
            T_ _ = TM„ _ . Therefore, they have the same cumulative
            nl’nc nl’nc
            distribution function and thus their asymptotic
            distributions are the same. There is no need to prove them
            separately.
            Theorem 3.4.1: Conditional on N, = n, and N„ = n , under H
            11 c c ’ o
            nl’nc
            - E ( T
            ni’V
            °(Tn n )
            nl>nc
            N(0,1) as n^ + °° an^ nc +
            where

            74
            E ( T
            nl’nc) = (Llnl(nl+1) + L2nc(nc+1))/4
            and
            a(Tn n ) = [Va r(T
            nl’nc nl’n,
            ))l,2
            = [(L^n1(n1 + l)(2n1 + l) + L2nc ( nc + l ) ( 2 nc+l ) ) / 2 4 ]l/2
            Proof:
            First, it will be shown that T and T have asymptotic
            nl nc
            normal distributions. Without loss of generality, it will
            be assumed that p = 0.
            Note that
            y ?. r . =
            . , i i
            i = l
            I n|x2i| - |xldL I) + l l V( IX211 - |x1±| + |x2i
            i = l
            where
            and
            2 , n
            1 < i < j <; n
            2jI - lXlj
            1
            - „!> +
            n,
            Di’ni' "Vi-iT(|X21^'|Xi1^
            ! " “ST J. .1 T(lX2i
            (,)
            l', 1 < i < j < n
            X1i I + lX2j I lXlj I }
            1
            are two U-statistics (Randles and Wolfe, 1979, page 83). It
            f o 1lows
            , a/2
            —— (T
            (/)
            3/2
            (n )
            1)/4) â– 
            (2‘) ‘ 1
            + (
            nl)1/2(U2,nE(U2,n>)

            75
            Now notice, 0 < U. <1 and under H„, E(Ui _ ) =
            i , n i 1 >n ^
            P { ( I X2 i I " |xlih > °> = V2, so that lUl,ni “ X/2| < V2 •
            Theref ore ,
            (n, )
            3/2
            (n . )
            3/2
            — (U - V2) < 7 pr ♦
            n j '-l,n1 z; n (n - 1 )
            0 as n * 00 .
            (/)
            (n,)V2
            Thus, (T - n (n.+ 1 ) / 4) and (n.)^n - V2) have
            (2‘) 1 1
            the same limiting distribution as n -► °° .
            By Theorem 3.3.13 of Randles and Wolfe (1979), it is
            seen that (n )^n - l^j has a limiting normal distribution
            1 z , n ^
            o
            with mean 0 and variance r E, ^ (provided £ ^ > 0)
            where
            r2?1 = 2 2{E[Â¥( |x2i
            Xlil + IX2 j I - lXljP
            x ?(|X2±| | x i i | + | X2 k I lXlklP ^4 >
            = 1/3
            Thus ,
            T - n.(n.+ 1)/4
            nl 1 1 d
            U‘)
            + N(0,1) .
            Note that,
            i ,v2
            U1) W
            1
            n (n + 1)(2n + 1 ) ,
            _ - 1 j fz
            P
            —+ 1
            24

            76
            as n1 > * . Therefore (after applying Slutsky's Theorem
            (Theorem 3.2.8, Randles and Wolfe, 1979)
            T - n (n + 1 ) /4
            1 1 1 d_
            a (T )
            nl
            N ( 0,1 ) .
            Similarly,
            n, +n
            1 c
            n. +n
            1 c
            l r3 +
            “V’
            j-n,+l “ n ^ +1 < j T = V Y.Q.
            n • J J
            c J =n ^ +1
            l { Y j f ( c j - ck) + Yk»(ck
            = j)}
            nc(D3,n > + (2C)(u4,n > â– 
            where
            and
            U4,n„
            1
            n, +n
            1 c
            3 »ri = — . ¿ YJ
            c J =n^ +1
            1
            l l Yi'i'ic. - c. )
            f0C\ n.+1 l 9 J 1 1 C
            k) + Yk'i'(ck - Cj)}
            are two U-statistics. It follows that
            <"=>m <-c>3/2
            —E_ rT - - ' - > w '• i _ c
            n ^ n
            (2C)
            „c(„c+ DM) . — (0 - E(U ))
            (,c)
            + ) •

            77
            Note
            that 0 < n < 1 , so | n - V2I < V2 and thus
            * r* 9 C
            , ,3/2 , ,3/2
            (n ) (n )
            c t „ 1 / >, c
            — (u, - V2) < c ■
            n ^ 3 , n v n (n
            + 0 as n *
            w)
            c c
            1 ) c
            u> 1,
            Thus, — (T^ - n^(n^+ 1 ) / 4) and (nc)72(U^ ^ - V2) have
            n v n c c
            Uc)
            the same limiting distribution as n^> <*> .
            Again applying Theorem 3.3.13 of Randles and Wolfe
            (1979), it is seen that (n ) U. - Vo] has a limiting
            c v 4 , n "
            c 0
            normal distribution with mean 0 and variance r “ £ ^
            2
            (provided r ^^>0 ), where
            r2^ = 22{E[(Yji(Cj - Ck) + YkT(Ck - C.))
            x (rjUcj - ci) + yít(cí - C.))] - l/4 }
            = 22{E(Yj'i'(Cj-Ck) + YknCk-Cj))(YjnCj-Ci) + Y±T(C.-C. ) ) - V4} .
            By the independence of
            Yj and Cj (Lemma
            r2^ = 22{P(Yj =
            + p(Yj- i)p(Y;L =
            + P(Yk= 1 )P(Yj =
            1 ) P(Cj > ck,
            1)P(C > C
            3 *
            DP(Ck> C ,
            3.2.2),
            C.> C. )
            J 1
            c.> c.)
            1 J
            c.> c.)
            J 1

            78
            + P(yk= 1)P(Y.= l)P(Ck> Cj, C.> C.) - V4 }
            = 4 {Vo P ( C . > C. , C . > C . ) + Va P ( C . > C. , C . > C . )
            z J k j i H j k i j
            V4 p(ck> c., c.> C.) +V4P(ck> c., c.> C.) - V4 }
            - ¿Í I x ir I x I x I 1 . V = I
            Thus ,
            T - n (n + 1 )/4
            n c c ,
            c d
            Vc) (3M14
            + N(0,1) . Noting that
            n f 1 nVo
            Uc) (^7>
            n (n + 1 ) ( 2 n + 1 )
            c c C nVo
            —■+ 1 as n ■>
            c
            24
            and applying Slutsky's Theorem, it follows that
            T -n(n+l)/4
            n c c ,
            - -+ N(0,1 ) .
            a ( T )
            n
            c
            The conclusion of Theorem 3.4.1 then follows by writing
            T - E(T ) (L.T + L T ) - (L E(T ) + L„E(T ))
            n, ,n n, ,n In, 2 n In, 2 n
            l cl c 1 c 1 c
            a(T )
            nl’nc
            (L2 a2 ( T ) + I2 a2 ( T )),2
            1 n, 2 n
            1 c
            L. a (T ) T - E (T )
            1 n ^ n n ^
            L„a(T ) T - E(T )
            L n n n
            c c c
            a(T n )
            nl’nc
            o ( T )
            nl
            a(T )
            nl'nc
            a ( T )
            n
            c
            x
            X

            79
            applying Slutsky's Theorem and utilizing the fact T and
            Tn are independent, conditional on = n-^ and N£ ~c
            = n
            â–¡
            Next, and most importantly, the unconditional
            asymptotic normality of T will be established as n
            n 1 ’ n c
            tends to infinity in Theorem 3.4.4. Prior to proving this,
            several preliminary results will be stated which are
            necessary. These preliminary results which are stated in
            Lemmas 3.4.2 and 3.4.3, were proved by Popovich (1983) and
            thus will be stated without proof. Minor notational changes
            are made in the restatement of his results to accommodate
            the notation in this dissertation.
            The first preliminary result, Lemma 3.4.2, is a
            generalization of Theorem 1 of Anscombe (1952).
            Lemma 3.4.2: Let {T _ } for n.=l,2,..., n =1,2,..., be
            n 1 >n c 1
            any array of random variables satisfying conditions (i) and
            ( ii ) .
            Condition (i): There exists a real number y, an array
            of positive numbers {cun n } and a distribution function
            F ( • ) such that
            lim P { T - y < x a) } = F (x)
            min(n^ ,n )-*■» 1 c 1 c
            at every continuity point of F(«).
            Condition (ii): Given any e > 0 and n > 0, there
            exists v = v(e,n) and d = d(e , n) such that whenever
            min(n^,nc) > v, then

            80
            p{ T , ,-T < e m for all n ,n' such that
            I n ' n n,,n' n,,n 1 c
            1 ’ c 1 c 1 c
            In' - n,| < dn., |n’ - n | 1 - n .
            I 1 1 I 1 1 c c' c
            Let (nr} be an increasing sequence of positive integers
            tending to infinity and let {N^r} and {Ncr} be random
            variables taking on positive integer values such that
            N. p
            i IT r
            > X . as r > , for some X. such that 0 l i i
            n
            r
            i=l,c. Then at every continuity point x of ?(•)
            lim P{Tn
            r >oo 1 r
            N
            cr
            y < xto , , r , , }
            [Xinr],[Xcnc]
            F (x)
            where [a] denotes the greatest integer less than or equal to
            a .
            Proof :
            This is Lemma 3.3.1 in Popovich (1983). Q
            The last preliminary result necessary is a result of
            Sproule (1974) which is also stated in Popovich (1983) as
            Lemma 3.3.3. It can be viewed as the extension of the well
            known one sample U-statistic Theorem (Hoeffding, 1948) but
            with the sample size as a random variable.

            81
            Lemma 3.4.3: Suppose that
            U
            n
            where B is the set of all subsets of r integers chosen
            without replacement from the set of integers {l,2,...,n} and
            f(t ^ , t2>..., t ) is some function symmetric in its r
            arguments. This Un is a U-statistic of degree r with a
            symmetric kernel f(»). Let {nr} be an increasing sequence
            of positive integers tending to infinity as r > « and (Nr)
            be a sequence of random variables taking on positive integer
            2
            values with probability one. If E{f(X^, X2»»«.,X )} < °°
            1/ o Nr P
            lim Var(nx2 U ) = r ? > 0, and 1 , then
            n 1 n
            n+°° r
            r \ r2^)^ } = Mx) ,
            lim P{(UN - E(Un )) < N
            r-)-® r r
            where $ (• ) represents the c.d.f. of a standard normal random
            variable .
            Proof : This is Lemma 3.3.3 in Popovich ( 1983 ). ^
            One comment is needed about this result. The proof of this
            lemma follows as a result of verifying that conditions
            and C2 of Anscombe (1952) are valid and applying Theorem 1
            of Anscombe (1952). Condition is valid under the null
            hypothesis and the verification of condition C2 is contained
            in the proof of Theorem 6 by Sproule (1974). This condition

            82
            C 2 will be utilized in the proof of the major theorem of
            this section which follows.
            Theorem 3.4.4: Under Hq ,
            \ ,Nc ' E(TN, ,N
            a(TN,,N }
            1 c
            — N(0,1) as n * °°
            Proof:
            The proof which follows is very similar to the proof of
            Theorem 3.3.4 in Popovich (1983).
            Let T
            nl’nc
            T - E (T )
            nl ,nc nl ,nc
            o(T )
            nl’nc
            , the standardized
            T statistic. Theorem 3.4.1 shows that {T } for
            n1» nc nl’nc
            n^=l,2,..., n =1,2,..., satisfies condition (i) of Lemma
            3.4.2 with y = 0 amd A5, it can be seen that X^ > 0 for at least one i=l,c. If
            X^ = 0, for i=l or i=c, then Theorem 3.4.4 follows directly
            from Theorem 1 of Anscombe (1952) and Lemma 3.4.3. Thus, it
            will be assumed that X ^ > 0 for i=l,c. The proof of
            Theorem 3.4.4 follows if it can be shown that condition (ii)
            of Theorem 3.4.2 is satisfied.
            Let T
            : - E (T )
            nl ni
            a (T )
            nl
            the standardized T
            n,
            statistic
            In the proof of Theorem 3.4.1, it was shown that

            83
            T has a limiting standard normal distribution by utilizing
            nl
            the U-statistic representation of T
            1
            As a result of Lemma
            3.4.3 and this U-statistic representation, it follows that
            T satisfies condition C., of Anscombe ( 1 952) (since T is
            n 1 L nl
            equivalent to a U-statistic which satisfies condition of
            Anscombe (1952) as proved by Sproule (1974)). This
            condition can be stated as follows.
            Condition C2: for a given e^> 0 and n > 0, there
            exists Vj and d^> 0 such that for any n^>
            * *
            T - T ,
            n 1 nl
            < e for all nj such that |n|
            - n,
            < d1n1 } >
            1 - n
            (3.4.1)
            Similarly, as a result of the U-statistic representation of
            Tn (as shown in the proof of Theorem 3.4.1) and from Lemma
            c
            3.4.3, it follows that T
            T - E (T )
            n n
            c a (T )
            n
            c
            satisfies
            condition C2 of Anscombe (1952). That is, for a
            given z0 and n > 0, there exists V2 and d2> 0 such that
            for any n^>
            P{ T - T . < e„ for all n such that n
            in n I 2 c I c
            c c
            n < d„n } >
            cl 2 c
            1 - n •
            (3.4.2)

            84
            Consider
            nl’nc
            - E(T
            nl’nc
            nl*nc
            °(Tn n }
            nl’nc
            L.o(T )
            1 nl
            a ( T )
            nl’nc
            * L2°(Tn >
            ) + — *
            1
            a (T )
            nl»nc
            K )
            Note that,
            (1) L^n and L2n ate functions only of N^ and Nc and the
            given and L2 constants.
            (2) (Lln)2 + (L2n)2 = 1•
            I ! ! P I
            (3) There exists constants and 1^ such that *â–º L^
            . P .
            and L„ *■ L„ as n + °° .
            7 n 7
            First, it will be shown that condition (ii) is
            f * f -Jc »
            satisfied for L í T ) + L„ÍT ) = T
            1 ^ n, ; 2 ^ n -1 n , , n
            1 c 1 c
            Let e > 0 and n > 0 be given and let v^, V2» ^1» ^2
            satisify (3.4.1) and (3.4.2). Let v = raax(vp V2) and
            d = min(d^, d2). Now,
            P { T , , - T < 2 e for all n , n such that
            In' ,n' n. ,n I 1 c
            1 c 1 c
            < dn. ,i=l,c)
            1 1 1
            1 *
            * 1
            ' 1 *
            * 1 -i
            T , â– 
            - T 1
            + L J T ,
            - T 1
            1 n'
            n, 1
            2 1 n '
            n 1 1
            < 2 e for all n', n'
            1 c

            35
            such
            that | n
            »
            i
            n.
            i 1
            < dn ¿, i = l,c}
            *
            V -
            1
            * i
            T i
            nl
            < t and
            f .
            l2I
            *
            V
            c
            * ,
            - T < e f o r
            n i
            c
            all nj,
            n'
            c
            such
            that | n
            t _
            i
            n .
            i i
            < dn., i=l,c}
            i
            *
            * 1
            T !
            nl
            < e for
            all
            ni
            such that |n J
            - nj <
            dn 1 }
            *
            ' n ’ "
            c
            * i
            T
            n i
            c
            < e for
            all
            n’
            c
            such that n’
            i c
            - n I <
            c 1
            dn }
            c
            i *
            lTn;
            *
            - T
            n,
            ¡ < e or
            !
            l2I
            *
            T ,
            n
            * i
            - T < e f or
            n i
            all n’,
            n '
            c
            c c
            such that n' - n, < dn, and n' - n < dn }
            11 II 1 I c cl c
            f . £ £ i i i
            > P{L. T , - T < e for all n' such that n' - n. < dn.}
            II n j n^l 1 II II 1
            1 I * * I I I
            + P{L„ T , - T < e for all n' such that n' - n < dn } - 1
            2 I n 1 n
            c c
            c c 1 c
            (3.4.3)
            Now using inequalities (3.4.1) and (3.4.2) and applying them
            to (3.4.3) with e = min{e^(L^) ^ ^ } then
            P T . . - T < 2e for all n,, n such that
            1' n,,n' n,,nl 1 c
            1 c 1 c
            nI _ nil < dni > 1 = 1>c} > (1 ~ h) + (1 - q) -1 = 1 - 2q

            86
            Therefore T satisifies condition (ii) of Lemma 3.4.2 so
            nl *nc
            that Theorem 3.4.4 is valid for T_ „ = L,T„ + L0T.
            nl ’nc
            In. 2 n '
            i c
            To see that the Theorem is valid if and L2 are replaced
            » I
            ^ and respectively, consider,
            nl >nc
            Tni»nc ' Lln Tnl
            c
            ' *
            - LlTn
            ' *
            L9t1 T„
            2 n n,
            +
            ' *
            L2Tn,
            = (L
            1 n
            - V1*, + (3.4.4)
            "fe
            Now, since Tn and TQ converge in distribution to standard
            normal random variables, Tn and Tn are Op(l) (Serfling,
            1 n c
            I ^ T
            ( 1980 ), pg. 8). Also, since L^n —■>• L ^ and
            * P * 11 11
            L 2 n + L2 as n > °°, (Lln - L ^) and (L2n - L2) are o (1).
            Therefore (3.4.4) shows that
            (L, - L,)T* + (L0 - L~)T* is o (1) and thus, Theorem
            In I n j 2n L nc p
            3.4.4 is valid.
            â–¡
            3.5 Comme n t s
            From the results in Sections 3.2, 3.3,and 3.4, it is
            clear that a distribution-free test of the null hypothesis
            of bivariate symmetry versus the alternatives presented
            could be based on T _ (or TM„ _ ). For small samples,
            nl»nc nl’nc
            an exact test utilizing the distribution of T (and
            n 1 ,nc
            TM
            „ „ ) conditional on N, = n,
            ,nQ' 1 1
            and N_ = n could be

            87
            performed. For larger samples, the asymptotic normality of
            T (and TM_ ) could be used. In Chapter Five, a
            n1’nc nl’nc
            Monte Carlo study will be presented which compares the CD
            test with the two tests presented in this chapter. For
            each, the asymptotic distribution will be used for samples
            of size 25 and 40 to investigate how the statistics compare
            under the null and alternative hypotheses for various
            distributions. First though, we make some comments on this
            chapter.
            Comment 1
            In Section 3.2, the test statistic Tn n , conditional
            on n£, was presented which had a null distribution
            equivalent to the Wilcoxon signed rank statistic. If
            instead of conditioning on nc, the statistic had been
            presented (with some minor adjustments) conditional on n2
            and n^, the statistic would then have had a null
            distribution equivalent to the Wilcoxon rank sum
            statistic. Conditioning on n£ and not on n2 and n^ was
            chosen because the observation of a particular n2 and n^ in
            itself, seemed important. That is, if only type 3 pairs had
            occurred (ignoring the number of type 4 pairs) that was
            significant, since under the null hypothesis, the
            probability a bivariate pair is type 3 is equal to the
            probabiltiy the pair is type 2. The signed rank statistic
            incorporates this idea and thus was used.

            88
            Comment 2
            In Section 3.3, the Kaplan-Meier estimate of the
            survival distribution was used in estimating the common
            location parameter. The usual median estimator (the sample
            median) could not be used, because in the presence of right
            censoring this estimator is negatively biased. Thus, the
            "smoothed" estimator based on the Kaplan-Meier estimate of
            the survival distribution was the logical choice.
            Comment 3
            The tests presented in this chapter are not recommended
            for situations in which heavy censoring occurs early on,
            that is, a lot of censoring in the smaller measurements. If
            this heavy censoring was to occur, many type 4 pairs would
            be present in the sample which are not used in the
            calculation of the test statistic other than to estimate the
            common location parameter. This test was more designed for
            situations when the extreme values (i.e., the larger values)
            tended to get censored.
            Comment 4
            In this chapter, statistics were presented to test for
            differences in scale when (1) the common location parameter

            89
            was known or (2) the common location parameter was
            unknown. The next natural extension would be to test the
            null hypothesis of bivariate symmetry versus the alternative
            that differences in scale existed with unknown location
            parameters which could be potentially different. This idea
            could be incorporated into the test statistic by using
            separate "smoothed" estimators for and This idea
            will be further investigated in Chapter Four.

            CHAPTER FOUR
            A TEST FOR BIVARIATE SYMMETRY VERSUS
            LOCATION/SCALE ALTERNATIVES
            4.1 Introduction
            In Chapters Two and Three test statistics were
            presented to test the null hypothesis of bivariate symmetry
            versus the alternative hypothesis that the marginal
            distributions differed in their scale parameter. This
            chapter will consider a test for the more general
            alternative, that is, that the marginal distributions differ
            in location and/or scale. To do this, two statistics will
            be made the components of a 2-vector, W , of test
            statistics. The first statistic denoted TE_ is a
            n 1 ’ nc
            statistic which is used to detect location differences. It
            was introduced by Popovich (1983) and is somewhat similar to
            the statistic introduced in Chapter Three. The second
            component of the 2-vector will be a statistic(s) which is
            designed to test for scale differences. Three different
            statistics will be considered for this second component.
            They are (1) TM (Chapter Three, Section 3.3), (2)
            nl’nc
            TM but using separate location estimates for X and
            n 1 y n c ^
            X2^ and (3) the CD statistic (Chapter Two). It will be
            shown in Sections 4.2 and 4.3 that the distribution of Wn is
            90

            on
            not distribution free, even when K is true. Thus, if f is
            o ’ rn
            the variance-covariance of W , the quadratric form W'
            ~ n ’ n ~nrn~n
            will not be distribution-free. A consistent estimator of
            A
            j] n» j-n t>e introduced in Section 4.5 and a test based
            the asymptotic distribution-free statistic W' i - W will be
            recommended for large sample sizes. For small sample sizes
            a permutation test will be recommended. First though, we
            introduce the TE statistic by Popovich (1983) with a slight
            change in notation to accommodate this thesis.
            Let
            Di
            = Xli
            - x2i
            a nd
            R( |
            Dil)
            be
            the
            absolute
            Di
            for i
            = 1,2
            > • • • >
            n, that
            is ,
            R(
            1 Di 1
            is
            the
            rank of |
            among (|
            Dll>
            1 ^2 1 ’
            • • • , |Dn
            | ).
            Define
            1
            if
            Z .
            ^ 0
            T .
            1
            = ¥(D1)
            = «<
            1
            0
            if
            Z .
            1
            O
            V
            •
            Let
            TE
            nl
            a nd
            TEn
            c
            be defined
            to
            be
            t he
            following:
            nl
            TE
            "l
            i
            l
            = 1
            4'1 R (
            lDi
            )
            and
            TEn = N - N2 .
            c
            Notice that TE
            is the Wilcoxon signed rank statistic
            applied to the n^ totally uncensored pairs. Popovich (1983)
            showed under Hq, N^ is distributed as a Binomial random
            variable with parameters nc and p = V2 ?2(0) = V2 P(type 2 or
            3 pair). With a slight modification from Popovich, the
            nl’n(
            statistic TE
            is

            92
            where
            + K2n
            TE
            TE - n. (n, + 1 )/4
            n ^ 1 1
            1 (n^r^ + 1 ) ( 2n L + l)/24)1/2
            and
            TE
            TE
            "c (
            (n ) z
            c
            and K^n and K2n are a sequence of random variables
            satisfying:
            1) and I<2n are only functions of and Nc,
            2) there exists finite constants and such
            P P
            that Kjn + Kj and ^2n * as n -»• °°.
            This is slightly different from the statistic Popovich
            introduced, the difference being that he required
            Kjn = (l~K2n) which is not being required here.
            One comment before proceeding to Section 4.2. In this
            Chapter, type 4 pairs will be ignored (except in estimating
            the location parameter for the scale statistics). This has
            no real affect since TE_ n > TMn and CD are not
            ni » nc nl ’ c
            affected by their presence (other than in estimating the
            location parameter). It will be assumed that the sample is
            of size n

            93
            4.2 The Wn Statistic Using
            c
            The first statistic to be considered for pairing with
            TE „ is similar to the statistic Tti _ presented in
            nl,nc nj,nc
            Section 3.3. The difference being, that instead of using a
            common estimate for p as in Section 3.3, here we first
            consider using separate estimates which are denoted by
            and M2 where Mj^ is the Kaplan-Meier estimate for p based on
            the X^'s alone and similarly, M2 is the Kaplan-Meier
            estimate for p based on the X2^'s. Define
            n
            = I V R(
            1 i = l
            and
            n, +n
            1 c
            3=^+1
            Y. Q.
            3 3
            (Note these are similar to statistics defined in Section
            3.2, with a slight modification of using the separate
            estimators Mj and M2») Similarly, define
            and
            T ~ n (n +1)/4
            * nil
            T =
            nl n^ ( n^+1)(2n^+1)/24
            T - n (n + 1 ) / 4
            n c c
            c
            n (n +1)(2n +1)/24
            c c c

            94
            which would be the standardized versions of T and T in
            nl nc
            Section 3.3, had not and M2 been used. We now defined
            / K TE*
            In n
            W
            1 n
            + K„ TE
            , 2n n
            1 c
            \
            * *
            L. T + L 0 T
            Inn, 2n n
            1 c
            where L^n and i,2n are a sequence of random variables
            satisfying:
            1) L^n and L,2n are functions only of and Nc
            and
            2) there exists finite constants and such that
            'In
            --*• and L2n
            -> L2 as n > .
            Note, the statistic for scale L ^ n T n + L 2 n T n is slightly
            1 c
            different than the forms presented in Chapter Three. The
            difference is that here, the two components are standardized
            before taking the linear combination. Appropriate weighting
            variables can be chosen though, which make this form of the
            statistic equivalent to that presented in Chapter Three.
            The test statistic for the alternative of differences
            in location and/or scale is wln where | \ is the
            variance-covariance matrix for W^n. The derivation of the
            asymptotic distribution of W ^ ' í'l”^ Wln’ wil1 be
            accomplished in a series of proofs. Theorem 4.2.1 shows
            that under H , if the common location parameter was known
            and used instead of the estimators M^ and M2, the vector
            T = (TE ,T (g),TE ,T )" has a limiting multivariate
            n1 nl nc nc
            normal distribution. Here T (p) denotes the statistic T

            95
            when the value of u is used in its calculation. T (u) is
            n ^ K
            the standardized Tn which was presented in Section 3.2.
            Next, Theorem 4.2.3 will prove that using the estimates,
            and M2 for the common location parameter, does not affect
            the asymptotic results in Theorem 4.2.1. This is achieved
            by applying results about U-statistics with estimated
            parameters (Randles, 1982) which are stated in Theorem
            4.2.2. Finally, the asymptotic distribution of Wj' W^n
            will be presented in Theorem 4.2.5.
            Theorem 4.2.1: When p is known and used in calculating T*
            under H and conditional on N^n, and N =n ,
            o 1 1 c c *
            T =
            TE
            „ni
            Tn1(“>
            *
            TE
            N ( 2 , t T > *
            where is the 4x4 variance-covariance matrix
            for T with
            oI<1>1) - ot(2’2> - ot<3-3> - 0l<4.4> - 1 ,
            = 12P (as defined on page 99) ,
            aT(3’4) = -(3/4)1/2 ,
            a (1,3) = (2,3) = a (1,4) = 0 (2,4) = 0 .
            Proof :
            Recall in the proof of Theorem 3.4.1., it was shown
            that

            96
            and
            Tn/W) = (nl>/2 O) (U2>ni- V2 ) + op(l)
            T* = (n „ )^2 (3)1/2 (U4 - V2 ) + on(l) ,
            '~c
            where U
            2 , n,
            1
            (nl)
            2
            I I *<|x21- >*| -
            1 Xli~ H + lX2j_ yl " lXlj
            1
            and
            U
            1
            4’nc ^c-, n.+1 < j U J 1 1 c
            l l ck) + Ykli'(ck- c.)}
            Similarly, Popovich's statistics can be written as
            TE
            ,v,,, !'i
            (UI>ni- V2 ) + opU)
            a nd
            where
            TE
            n
            V,
            (nc> U3,n
            U
            1 , n
            1 (nl) l 2
            I l T(XU- X21+ Xj.- X2.)
            nl +nc
            U, = — l (1“2y . )
            3 , n n L . j
            c c i-i^+1
            and

            97
            Thus ,
            TE
            n
            1
            * / \
            TEÍ
            c
            T‘c i
            (r,;)1^ (3 )l/2 (U1>ni- V2 )
            <”i>1/2 ojVa (u2 ni- V2 )
            <"c)1/2 <°3,nc>
            (nc)V2 C3)V2 (U4>nc- V2 )
            op(l)
            °p(l))
            op(l)
            and therefore, if we can show the right hand side has the
            appropriate distribution, the proof will be complete.
            First, it will be shown that
            U =
            where
            °1/2
            "/2
            1,1/2
            c
            —> N(0,i )
            - T u
            "/2 (D4,nc- V2 )
            , , (a , b) ^ , ( a , b ) v 1
            = ((a )) and a = ¿ 7
            2
            I
            i = l
            rU)r(b)
            ( a , b )
            lim (—) and £ ’ is the covariance term described in
            n-*"=° n i
            Theorem 3.6.9 of Randles and Wolfe ( 1979 , pg . 107).
            Note, conditional on N^=n^ and Nc=nc, the problem can be
            considered as a two sample problem. By Theorem 3.6.9
            (Randles and Wolfe, 1979, pg. 107) it follows that

            98
            U =
            "1/2 \ >

            A (U3,nc)
            "‘/2
            7 P
            —-> N(0, ^u),
            where jlu = ((o^a,b^)) and
            (a , b ) =
            (a)(b)
            i i (a , b)
            i-i h 1
            2
            I
            for - lim (“~) and (rj3^,^^^) the degrees for
            n-»-<»
            U-statistic U . Here, U, and U9 are of degree (2,0),
            a l,Llj z.,n^
            U3,n is °f degree (0,1) and n is of degree (0,2). We
            c * c
            now evaluate the matrix j;u<
            Now,
            (2,2) _ 2x2 .(2,2) 0x0 (2,2) _ 2x2 (2,2)
            h ?1 »2 C2 " S C1
            whe re
            ?1(2’2) = Cov{ 4<( |X21- \i | - |x1±- y
            1 + 1
            X2j- U
            1
            T-)
            r—i
            X
            1
            ^(1X2i~ H1 " |Xli- U
            + |
            X2k~ ^
            1 - lXlk-
            = E{T(|x2i- y| - |x1±- u| +
            lX2j-
            -1 -
            lxlj- w|)
            xn|x2i- u | - |xu- u | + |x2k-
            U I -
            1 xik-
            y | )} - V4
            Notice that under HQ , |x2^- d| _ |^ii~ l1) is symmetrically
            distributed about 0
            Thu s

            99
            C1(2,2) = 1/3 - 1/4 = 1/12
            and
            a ( 2 ’ 2 } = l/(3Xj) .
            Similarly,
            „(1,1) - 2x2. (1,1)
            X M
            where
            5l(1,1> ‘ Cov(T(Xu- X21+ X,r X2j), »(Xlt- X21+ Xlk- X2k)}
            = 1/12
            s o
            a (1 ’1) = l/OXj) .
            Likewise,
            a(3’3) = (1/X2)?2(3’3) = (l/X2)Cov{l-2Yi,1-2 Y±}
            = (l/X2)4xVar(y^) = 1 / X 2 ,
            a ( 4 ’ 4 ) = ( 4 / X 2 ) Cov { y ^ ) + Yj'1'(cj-Ci).
            Y i ^ ( c i ~ ck ) + V^k^i*}
            = 4/X2(l/12) = 1/(3 X 2 ) ,
            (1,2) _ 2x2 (1,2) 0x2 (1,2)
            ( 4/X j)Cov{¥( |x2^ p | — | X ^ ^ P j 4" | X 2 2— P | 1^12 P | )
            '{'(Jin- X21+ x13~ X 2 3 )}
            = (4/X1){Pr[( |X21- W| * |xn- p| + |X22- p| - |X12- p|) > 0,
            (x 11 — X21+ X13- x23) > 0] - V4 }

            and
            Thus ,
            where
            Define
            100
            s 4P*/X1 ,
            <3>4> - (2/*2H2(3'4)
            = (2/X2)Cov{ Yi'*'(ci_ck_) + Yk'l'(ck-c;L ) , 1 — 2 y ± }
            = -(4/A2){E[Y2f(ci-ck)+YiYk’l'(ck-ci)] - V4 }
            = -(4/X2){E[Yi'l'(ci-ck)]+E[YiYk'i'(ck-ci)] - V4}
            = ~ ( 4 / X 2 ) ( V4 + V2 x V2 x V2 - V4J
            (by Lemma 4.2.1)
            = -1/(2X 2 ) ,
            r ( 1 * 3 ) = 0(2,3) = o.
            we have
            "/2 OJl.nj- lk >
            "V2 (»2,n,- *4 >
            "/2 <03,nc)
            A (“4,n -*4 >
            — * N(0, |u),
            ♦ u"
            1/(3 X x )
            (4/X1)P*
            0
            0
            (4/Xj)P*
            l/OXj)
            0
            0
            0
            0
            1 / X 2
            â–  1 / ( 2 X 2 )
            0
            0
            -1/(2X2)
            1 / ( 3 X 2 )
            a 4x4 matrix A to be,
            A =
            (3X^2
            0
            0
            0
            ( 3 X L )f2
            0
            0
            0
            0
            ( X 2 )*'2
            0
            0
            0
            0
            ( 3 X 2 )1/2

            101
            and applying Corollary 1.7.1 in Serfling ( 1980 , pg . 25), it
            f o1lows
            'i Vo ( o \Vo
            A U =
            ( n X i ) ' 2 ( 3 )' 2 (U1>ni- V2 )
            (nAj)1^ (3)V2 (U2>n - V2 )
            (nX2)V2 (U3,nc>
            (nX2)1/2 ( 3 )X/2 (U4 jn - V2 )
            —> n( o, j:T)
            where
            I* T A|-uA'
            1
            1 2 P*
            0
            0
            1 2P*
            1
            0
            0
            0
            0
            1
            â– (3/4)1/2
            0
            0
            - ( 3 / 4 )l/2
            Recalling, that X^= lim —^ , the proof of Theorem 4.2.1
            n -*â– <*>
            follows.
            â–¡
            The next step in proving the asymptotic distribution of
            Wi¿ j: J * ^ln » be to show that the estimators and M2
            do not affect the asymptotic distribution of T. Theorem
            4.2.2, states results about U-statistics with estimated
            parameters (Randles, 1982). This theorem condenses
            Conditions 2.2, 2.3, 2.9A, Lemma 2.6 and Theorem 2.13 from
            Randles (1982) into the statement of Theorem 4.2.2.

            102
            Theorem 4.2.2: Given the following three conditions
            1)Assume there exists a B^>0 such that
            < for every
            Xj x and all y in some neighborhood of p , where
            h( • ;t) denotes the kernal of the U-statistic Un.
            2)Suppose there is a neighborhood of X, call it K(X)
            and a constant B2>0 such that if yeK(X) and
            D(y,d) is a sphere centered at y with radius d
            satisfying D(y,d) c K(X) then
            E [ Sup |h(X
            y ' e D ( y , d )
            xr;T)|]
            .,Xr;y') - h(Xx
            > • • • y
            (Condition 2.3)
            3)Assume [h(X^,...,Xr;y)] has a zero differential
            at y =y, that
            and
            where
            a
            2 = Var{E[h(X1,...,Xr;u)|xi]} > 0
            (Condition 2 . 9A)
            then
            n/2 [U (i) - U (y ) ] —> 0
            L n n J
            Proof:
            See Randles (1982).
            â–¡

            103
            Theorem 4.2.3: Under HQ ,
            (n fa [T* (y) - T* (M.,M.)] 0
            1 nl n ^ 1 2 J
            tic
            where Tn (y) is the statistic Tn which used y, while
            *
            Tn is the statistic which use estimates of y,
            and .
            Proof:
            The proof of Theorem 4.2.3. follows, if the conditions
            of Theorem 4.2.2 hold. Although Theorem 4.2.2 has been
            stated here in terms of one parameter, the theorem is valid
            for a p-vector parameter (i.e., y). Thus, the unknown
            parameter in this case is (y^,y2) which is being estimated
            by (M^,M2). Next we need to show that the necessary
            conditions hold.
            Note, Condition 1 follows directly from the fact that
            the kernel for Tn is an indicator function, that is
            'P ( t) =
            1 t> 0
            0 t<0
            and thus
            |n|x2i-r2! - IXji-Yil + |x2J-Y2| - | Xj j - Y! | )
            - Ixlj-u|>j
            - ’1'(|x2i~>J| “ |xli-p| + | x2 j
            < 1.

            104
            To prove Condition 2 holds, it needs to be shown that
            E[ SUP I^(IX2i“Y¿| ~ |xií-y1I + IX2j-^2I ” IX1j | )
            I'cDCj.d) J
            - 'F(|x2i-y2| - |xil-y1 | + |x2j-Y2| - | X i j — Y i | ) ] < B2d.
            (4.2.1)
            First, consider the following change of variables, let
            f f
            Yi i = X,£- yi and Y2i = X2i~ Y2* This simplifies (4.2.1) to
            showing
            E[ sup |T(|Y21| + |Y2j| - |Y1±| - |Yjj|) -
            Y eD(y , d )
            t(|Y2i-(Y2-Y2>| + |Y2J-(Y2-Y2)|-|’ili- < B 2d
            (4.2.2)
            Recalling that | T(t) - ¥(t1 ) j < 1, we need to show
            t’r[|*(|Y21l + I Y2j | - |Ylt| - | Y j j | ) -
            »(|Y2í-(Y2-Y2>My2j-|-|Yií-Hy1:)_(Yi-t;)|)|
            = 1 ] < B 2 d
            (4.2.3)

            105
            To prove (4.2.3) we will utilize the fact that | Y 2— Y 2 I * ^
            ¡ti 1
            and IY 2 ~ ^ 2 I * ^ (i*e., y cD(y,d)) and first consider the
            region where (y^I > d and | Y 2 ^ | > d for k=i,j. Without
            I
            loss of generality, it will be assumed that and
            f
            y 1 >y 1. It will be argued that this region can be
            appropriately bounded, and similarly that the regions which
            have not been included here can be bounded also.
            For the first region we are considering (i.e.,
            I^lkl ^ ^ an<* I Y 2 k I ^ d, k=i,j) notice that it can be
            divided into 16 subregions determined by |Y^(y^-y^)| and
            i f i
            I Y 2 ^— ( Y2~y2^ I h = i,j that is, determined by whether Y^ > d
            or < -d for k = i,j and whether Y2jc is > d or < -d for
            k=i,j. Consider the subregion where Y^> d, d, d
            and Y2j> d. In this region (4.2.3) simplifies to
            Pr[|m2i+ Y2J- Ylt- Yjj)
            - 'l'(Y2i+ Y2j- YU- Ylj- 2 + 2CYl-r;>)| • 1] .
            (4.2.4)
            Letting Y = Y2^ + Y2j - Y^ - Y^j, (4.2.4) becomes
            P[IT(Y*) - T((Y*- 2(y2-Y2) + 2(Yl-yJ))| = 1]
            which is equal to
            2(y 2“Y 2) ~ 2(y1~y1)
            f(y )dy
            if 2(y 2-y £) - 2(y1~y1) > 0
            or

            106
            ( * *
            J f (y )dy
            2(Y 2 ~Y 2 ) ~ 2(y1”Y1)
            if 2(y2"Y2) “ 2(y1“Y1) < 0 ,
            •jlf "ft
            where f(y ) denotes the density function of Y . Now f(y )
            is bounded, if X^ and X2^ have a bounded joint density.
            Letting this bound be denoted by B (finite), then
            2(y2“Y2) " 2(Y1~Y1 )
            * *
            f(y )dy
            < B
            2(Y2-r2) - 2ÍYJ-Y,)
            I d(y)
            and similarly,
            < 4 B d
            u
            /
            f (y ) dy
            » t
            < 4Bd .
            2(y2-Y2) “ 2(y1“Y1)
            Thus, this subregion is bounded. With similar arguments,
            the remaining 15 subregions can be shown to be bounded with
            the same type of expression.
            Similarly, the other regions, (i.e., { | Y ^ ^¡ < d and
            < d, k=i,j}, etc.)
            can also be bounded by K^d for some constant . This
            completes the proof of Condition 2.
            Y2k| < d, k=i,j}, {|Ylk| > d and |Y2k
            For Condition 3, under the following conditions,
            oo oo u
            iff fv(u-s,u)f (v+s,v) ds du dv
            0 'o -v X X

            107
            0 0 -v
            f f (u-s,u)f (v+s,v) ds du dv
            * A A
            — OO —00 U
            and
            oo 0
            / /
            0 —»
            â– / /
            0 — oo
            V
            r f (v-s,v)f (-u+s,u) ds du dv
            A A
            u
            V
            J" f ( u-s , s ) f ( - v + s , v ) ds du dv
            u
            where f^(*,*) represents the density of
            (4.2.5)
            it can be shown that the differential is zero. For the next
            requirement of Condition 3, under certain regularity
            conditions, it can be shown that
            n^[M^~ p] = 0(1) for i=l,2 .
            The regularity conditions which are required are the
            following:
            a) that Fy (•) is continuous, where Fv is the
            xi *i
            /
            marginal c.d.f. for Xj^ i=l,2 ,
            b) that G(•) is continuous,
            and
            c) G(F~|( V2 )) < 1 .
            Note, conditions a and b are satisfied by assumptions A2 and
            A3 and that condition c requires the censoring distribution

            108
            to have support which includes the location parameters
            p^ and ^2 which is satisfied by assumption A6 . (See Sanders
            (1975).) Thus Condition 3 holds since
            n/2 [U (p) - E(U (n ) ] N( 0 , a2 )
            n n
            was shown is Section 3.2. and therefore the proof of Theorem
            4.2.3. is complete. Q
            Corollary 4.2.4: Under Hq ,
            l/„ * * P
            n 2[t (p) - TM ] * 0
            ni ni
            ^ £
            where TM„ denotes the T statistic which uses the combined
            nl nl
            sample estimate for p.
            Proof :
            This can be viewed as a special case of Theorem
            4.2.3. Note, in this case it is easily shown that
            E ^[h(X^ , . . .,X r;y)] has a zero differential by noting that
            |^ 2 i— ^I ~ |^li~Y| has a symmetrical distribution about 0 for
            any y. The extra conditions stated in (4.2.5) are not
            needed. lJ
            It has been shown in Theorem 4.2.3 (or Corollary 4.2.4)
            that using the estimates and (or M) does not affect
            the limiting multivariate normal distribution of the vector
            T of test statistics. The last major theorem of this
            section, Theorem 4.2.5., states the resulting asymptotic
            distribution of the quadratic form W£n ^ l n * t'ias

            109
            theorem T (M,,M9) will denote the T statistic using the
            n ji i l n. ^
            separate estimators and M2.
            Theorem 4.2.5; Under HQ , the following are true,
            ( 1 ) W
            1 n
            K. TE* + K„ TE*
            In n, 2 n n
            1 c
            L. T* (M. , M„ ) + L- T*
            Inn, 1 2 2nn
            1 c
            N(0, fj)
            where j: ^ = ( a ^ a ’ ^ ^ ) with
            al ( 1 ’ 1 ) = K2 + K2
            a^2’2) = L2 + L2
            a/1 >2) = 12K1L1P* - K2L2(3/4)1/2 ,
            and
            <2> “in I!' “in —'* X(2) .
            (3) if |^is any consistent estimator of j ^ , then
            "in ?T‘“ln —
            d 2
            (2)
            Proof :
            To prove part (1), note that from Theorem 4.2.3.,
            we
            have
            TE.
            T =
            Tn(Mi ,M2)
            TE
            n(o,|t) ,

            no
            where j;T was defined in Theorem 4.2.1. Defining a matrix A
            to be
            A
            and applying Theorem A in Serfling
            (1980, pg. 122) we see that
            Win = AT —* N(0, AfTA')
            whe r e
            AT =
            * *
            K, TE + K „ TE
            In. 2 n
            1 c
            L T* (M ,M ) + L T*
            In. I Z Z n
            1 c
            í i
            /
            Thus, part (1) follows by noting that
            K. TE* + K„ TE* - ÍK.TE* + K „ TE *
            In n. 2n n '-l n, 2 n
            1 c 1 c
            = f K. - k.Ite* + (k„ - k„)te*
            'â– In 1; n, v2n 2J n
            I (
            X x
            and since TE and TE„ converge in distribution to standard
            nl nc
            normal random variables, TEn and TEfl are 0 (1) (Serfling,
            1 p c p
            (1980), pg.8). Also, since (K. > K,)
            In 1
            P
            and (K^n * ) as n > <*>, thus (K^n~ K^) and (K2n~ K2^ are
            op(l). Therefore, (Kln> Kj)TE* + (K2n“ K2)TE* is op(l). A
            similar arguement holds for L^nTn (M,, M2 ) + L2nTn
            1 c
            and thus

            the vector W, has the same distribution as AT.
            -In
            Parts 2 and 3 follow directly from (1) and well known
            results.
            The results in Theorem 4.2.5 also hold for
            *2n
            * *
            R, TE + K0 TE
            In n, 2 n n
            1 c
            L, T* (M) + L„ T*
            In n
            1
            2n n
            where T (M) denotes the T statistic using the combined
            n 1 nl
            location estimate and thus will not be stated separately.
            The quadratic form based on W2n is denoted ^2n where
            11 = t 2 ‘ Note that each quadratic form mentioned in this
            chapter is not distribution-free, although each is
            asymptotically distribution-free. Section 4.5 will
            investigate consistent estimates for jl ^ .
            4.3 The W^n Statistic Using CD
            The last statistic to be considered for pairing with
            TE„ _ is the CD statistic presented in Chapter Two.
            n 1 ’ nc
            Here, the statistic will be denoted by , to indicate
            this third type of scale statistic used.
            Theorem 4.3.1: Conditional on N, = n. and N = n , and
            i 1 c c ’
            o ’
            under H

            112
            / K, TE + K„ TE
            / In n 2n n
            W,
            - 3n
            N(0 , |3)
            CD
            where j; 3 is the variance-covariance matrix for W-jn and CD
            is the standardized CD statistic of Chapter Two.
            Proof:
            Recall from Chapter Two (ignoring type 4 pairs), that
            CD =
            • n.+n ,
            1 CJ
            l l a . b . .
            i If you have only type 1,2, or 3 pairs, then there are three
            possibilities for the i^ and j1"*1 pair types, which are
            t-1_ t* V»
            1) the iLn and j pairs are both type l's (i.e.,
            uncensored, 6^=6j=l),
            2) the iC^ and j11*1 pairs are both type 2's or 3's
            (i . e . , (6^,6j)e(2,3) where this indicates that 6^
            and 6j are both elements of (2,3)),
            or
            3) the ith pair is type 1 and the pair is type 2 or
            3 or vice versa (i.e. {5^ = 1 and 6^e(2,3)}
            or {6^e(2,3) and 6^=1}) .
            Here, it will be assumed, as in Chapter Three, that the type

            113
            1 pairs occupy positions 1,2,. ..,n^ in the sample while the
            type 2 or 3 pairs occupy positions n^+1,n2+2,...,n^+nc.
            Thus, CD can be written as
            1
            CD = f { l l a b
            ,• n , +n >, 1 , L . .. L ij ij
            ( 1 c) l 1 I a b. .
            n +1 1 J c 1
            n , n,+n
            1 1 c
            + l l
            i=l j=n1+l
            a . . b . . }
            ij ij J
            which is in the form of three U-statistics, that is,
            CD =
            1
            n, +n
            ( \ c)
            2>ic + U>2c + “l* "c“3 = l
            where
            1 c n
            — I l a . b . .
            L L ij ij
            1 1 < i < j < n
            1
            U
            1
            l l
            2 c n ,,„ . . . ^
            , c \ n,+l lo J 1 c 1
            a . . b . .
            ij ij
            and
            n, n. +n
            l 11c
            U, = l l a b
            3c n.xn .L. .L ,, ijij
            1 c i = l j =n +1 J J
            J 1
            Note, that U^c and U2C are one sample U-statistics while
            is a two sample U-statistic.
            Using the fact that,
            *
            TE„ „ = K,TE* + K9 TE*
            n 1 »nc In n i 2 n n^
            where

            114
            Mi
            TE*, - (nj/4 (3)''2 (Ul>ni- V2 ) + or(l)
            1
            ll
            and
            v,
            '2
            and from Theorem 3.6.9. of Randles and Wolfe (1979, pg
            107), it follows that
            "/2 (“l.nj- \ >
            "/2 (U3,„C)
            nV2 n72 (U2c)
            nV2 (U3c)
            N ( 0 , Í )
            ' T u
            where jlu = and
            , (a) (b)
            (a,b)_ r ri ri _ (a,b)
            0 L 4-j
            i = l X.
            i
            Here U, _ and Ulo are U-statistics of degree (2,0), Uo
            I,U^ IC J y ll ^
            is of degree (0,1)., U 2 c is of degree (0,2) and U 3 c is of
            degree (1,1). From the proof of Theorem 2.4.1, we have
            a(1’1) = 1/(3X1), a(2,2) = 1/X2 and a(1,2) = 0. In
            addition,
            °(3’3) - (4/X^Covla^bjj .«ikbiklii-Sj-ik-ll.
            f(4»4) = ( 4 / x 2 ) C o v { a ^ j b^ j , a ^^b ^^ | ( 6 ^ , 6 j , 6 ^ ) e ( 2 , 3 ) } ,

            115
            0(5,5)
            = (1/X1)Cov{a±jb ±j
            ’aikbik1
            6i = 1’(6j .Sk)e(2,3)} ,
            + (l/X2)Cov{aikb.k
            »ajkbjkl
            6i = 6j = l,5ke(2,3)} ,
            „(3,4)
            - cl1’*) = o(2’3) =
            0,
            0(3,5)
            = (2 /X 1 )Cov{atj b¿j
            ’aikbik1
            6 ± = 6 j = 1,5 kc(2,3)} ,
            0(4,5)
            = (2/X2)Cov{a±^bij
            ’aikbik1
            (6i , 6j )e(2,3) ,6k = l} ,
            a(l,5) _ (2/Xj)Cov{V(¿"X2i+xij-x2j^» aikbikl
            6^=6j=l,6jcE(2,3)} ,
            a(i,3) m (4/X1)Cov('P(Xu-X2i+X1.-X2.), aikbikl 6i = 6j = 5k=1 1 ’
            a(2,5) _ (i/x2)Cov{l-2yj >aijb±j|<5¿ = 1 , 6je(2,3 ) } ,
            a(2,4) = (2/X2)Cov{ 1-2 Yj , ai j bi j | ( 6 i. <5 j ) £ ( 2,3 ) }
            Next, we get the distribution of
            l/~ Vo Vo
            (n (Ul,ni - l/2 >» a (U3,nc>* n (CD))'
            Note that,
            V2 (CD) , + (">2c + nincU3c} ,
            l 2 J

            116
            1 i m
            n-voo
            nl
            (/)
            (?)
            = li m
            n (n1-1 )
            n(n-1)
            = X.
            lim
            n-*-°°
            Í2Ü
            (?)
            = X.
            and
            lim
            n >a>
            n, n
            1 c
            (?)
            = lim
            n->"»
            2n, n
            1 c
            n(n-1)
            2 X 1 X 2
            Thus, we need
            1 0 0
            0 1 0
            0 o x2
            0
            0
            0
            0
            2X^2 /
            "/2 ("l.nj - lk >
            "/2 nV2 (0lc)
            "/2 <"2c)
            nk (U3c)
            "/2 V2 )

            m 2 it + v 2
            "/2 V2 )
            nV2 (U3in<;)
            nL/2 (CD)
            with variance-covariance matrix A j: A'
            where
            (1,1) = a(l,D
            f = ( (a,b)
            fCD ' °CD
            'CD
            ’CD
            (2,2) = (2,2)
            0 (1,3) = x2 (1,3) + 2X x -(1,5)
            aCD Ala + /A x A 2 o
            a (2,3) = x 2 ( 2 ,4) + 2X x (2,5)
            aCD A2a 2AlA2a
            and

            117
            °CD
            (3,3)
            = A^3 ,3) + 2A^(2XxX2
            + 2A2(2A1A2)a^4,5) + A
            + (2A:A2)2a^ 5 ’ 5 ^ .
            )a(3’5>
            2 a ^ 4 ’ 4 ^
            Next, it will be argued that a£D^3,3^ is asymptotically the
            same as the variance for CD (i.e., 4y) derived in Section
            2.4. Note that,
            2 (35) 2 (45) 22(55)
            2A (2A1A2)o'‘ ’ ;+ 2A2(2A1A2) a ' 4AXA2 a’
            = 8 A
            lX2Cov 2
            , a b ., I 6 .
            lk lk 1 1
            = 6 . =
            J
            1 ,6ke(2,3
            )}
            +
            8A2 A jCov{a
            ub
            ij ’
            aik
            bikl
            (6i,
            6 ) e ( 2,3 )
            ,6
            k = 1l
            +
            2 f
            4X ^ {a
            2
            ub
            ij ’
            aik
            bikl
            6i = 1
            ,(6 , 6k ) e
            (2
            , 3 ) }
            +
            4A ^ A2 Cov{a
            ikb
            ik *
            a .,
            jk
            b)J
            6i=6
            r1-5^2
            ,3
            )}
            = 4 A
            !A2 Cov{
            K
            = 6 .
            J
            = 1,6
            k£(2
            .3)1
            +
            2
            4A ^A 2 Cov{
            K-
            V
            !,6.
            e(2,
            3>1
            +
            2
            4A1A2 Cov{
            l(5i
            ’6J
            )e( 2
            ,3) ,
            V1!
            +
            2 .
            4A A2 Cov{
            l(6i
            ’6k
            )e(2
            ,3) ,
            5.-1)
            +
            2
            4A A2 Cov{
            K=
            1 ,(6 ,6
            k)e(2,3)}
            +
            2
            4A ^ A2 Cov{
            lv
            (2,
            3) ,6
            • = 6.
            J k
            = 1} *
            Thus, combining all the terms in o^jj^3,3^, we have
            lh-5j-6k'1l
            I < <5 i » <5-j ,5k)e(2,3)}
            I 6i = 6j=1 »6ke(2,3)}
            °cd(3’3) = 4XlCovi
            + 4A2Cov{
            2 .
            + 4A^A2Cov{

            118
            + 4A j A 2C0v{
            2
            + 4A j A 2 Cov{
            2
            + 4A^A2<"ov{
            , 2 ,
            + 4A ^ A 2 Cov{
            2
            + 4A ^A2Cov{
            Recalling that,
            6i=6k=1 ’6je
            (2
            ,3)}
            (6±,6j)e(2,
            3)
            ’6k =
            U
            (5i,6k)e(2,
            3)
            ’6i =
            1}
            6k=1>(6j’6k
            ) e
            (2,3
            )}
            61e(2,3) ,6j
            = 6
            k = 1l
            (4.3.1)
            n
            1
            n
            = (proportion of sample which are type l's)
            P
            â– +â–  A^ = (probability of being a type 1),
            and, thus, the A coefficients in front of each covariance
            term are the probabilities necessary to uncondition each
            covariance term. For example,
            3
            X1Cov(aljb1J , aikbikK-«j-«k-l>
            a ., b .,
            lk lk
            6i = 6j = 6k=D
            Cov(aijbij , a n. t b n. t ,6-j ~<5^ —6t — 1) .
            ik ik
            Now note that the eight covariance terms correspond to the
            eight possibilities for the subscripts i, j and k (i.e., 3
            subscripts with 2 possibilities for each, that is, each
            3
            subscript is either a 1 or (2,3) yields 2 =8 combinations)
            and thus
            'CD
            (3,3) = Cov(aij b^j »ai'jbi'j) = 4y
            The last step of the proof, (i.e. showing — > N(0,j: )
            W3
            follows from observing that

            119
            í KjOXj)1^
            \ °
            K2(X2)1/2
            0
            ^y)A/2
            \
            )
            "/2 (U3,nc
            n4 (CD)
            KjO)1^ (n X L )^2 (Uln - V2) + K2(nX2//2 (U
            3 » n.
            —* nco.L)
            CD
            by Theorem A in Serfling ( 1980 , pg . 122), where |3 =
            (a3(a’b))
            a3<2’2>
            and
            o^1’2’
            2 2
            Ki + K2 >
            1
            K1(3X1)/2 {xl°(1’3) + 2X^20^ ,5)} (4y)"1/2
            + K2(X2)1/2 {x2o(2,4) + 2X1X2a(2’5)}(4y)"1/2 .
            After some simplification, similar to that used to show that
            was equivalent to 4y, it follows that o3^’2^ is
            equal to
            V4^2 <3X1)1/2 (‘CovlTCX^- X21+ Xj.- X2.), alkblk|Si-6.-l]}
            + K (4y)_l/2(X )X/2 {2 Cov ( 1 -2 y a b | 6 e ( 2,3 ) ) } .

            120
            Recalling that,
            —— X, and —- —X„ , and using a similar
            N 1 N 2
            arguement as in Theorem 4.2.5 (pgs. 109-110), we get
            k k
            K, TE + K„ TE
            In n, 2n n
            1 c
            CD
            /
            N ( 0 , | )
            3
            and the proof of 4.3.1 is complete
            â–¡
            Note that, similar to the case with W^n and
            ?3n 131 W 2n is not distribution-free, although it is
            asymptotically distribution-free .
            The following corollary, states results which follows
            directly from Theorem 4.3.1.
            Corollary 4.3.2: Under H ,
            -1
            o
            d 2
            (1) -3n E3 -3n + x(2)
            and
            (2) is any consistent estimator of j: ^ , then
            " -1 d 2
            -3n Z3 - 3 "â–  x (2) *
            Proof :
            The proof is omitted, since (1) and (2) follow directly
            from Theorem 4.3.1 and well known results. O

            121
            This section has established the asymptotic
            1 ^3n w^ich could
            be used for a large sample test for the general alternative
            of location and/or scale differences. The next section,
            will discuss a permutation test which could be performed for
            any of the quadratic forms (based on W^n, W2n or W^n)
            mentioned in Sections 4.2 and 4.3. Section 4.5, will
            discuss consistent estimators for | ^ , jl2 and j: 2 .
            distribution for the quadratic
            form W^n
            4.4 Permutation Test
            In the situation where the sample size is small, there
            may not exist a good estimate for j; ^ i = 1 ,2,3 or for the
            2
            limiting X(2) distribution to provide an adequate
            approximation for the distribution of Win Win i = l,2 or
            3. In this case, a small sample permutation test is
            recommended.
            Recall, in Section 2.3, a permutation test for CD was
            discussed. It was based on the 2n possible samples
            {[Xii,x2i,6i)ki,(xi2,x22,52)k2,...,(xln,X2n,6n)kn] :
            = 0 or 1 for i=l,2,...,n}
            which are equally likely under HQ . Here

            1
            X2i’Si> ‘ ' '
            122
            if k.
            i
            if k.
            i
            The permutation test in this section is based on the same 2n
            samples which are equally likely under Hq . (A slight change
            is present though, since n = + Nc here, that is, no type
            4 pairs are included in the sample for the calculations.)
            Without loss of generality, it will be assumed that
            Wjn is the test statistic for which the permutation
            test is being done. The permutation tests for the
            statistics based on W2n and W^n are performed similarly. It
            is also assumed that a particular K^n and K2n (L^ and L2n)
            have been chosen by the researcher.
            Let win(l) denote the first component in W^n; that is
            the location statistic which is
            * *
            w. (1) = K. TE + K. TE
            In lnn^ znnc
            and let w¡n(2) denote the second component in W^n; that is
            the scale statistic used for w^n* For each of the 2n
            equally likely samples, the statistics Wjn(l), w¡n C 2) and
            wlnwin(2) are computed and their values tallied. From these
            tallies, the relative frequency of each possible value of
            w, (1), w, (2) and w. (l)w, (2) is determined and these
            relative frequencies are then the probabilities that Wjn(l),
            w^n(2) and wjn(1)w^n(2) assume the corresponding distinct
            values. Using these probabilities, the actual conditional
            variance of w^n(l) and w^n(2) can be calculated and the

            123
            actual conditional covariance of w,n(l) and wi2n^) can
            calculated.
            Let | denote the conditional variance-covariance
            matrix for W^n. Now we calculate the 2n not necessarily
            distinct values for W^n j;“* W^n determined by the 2n equally
            likely samples under Hq . From these calculations compute
            the relative frequency for each distinct value of
            Wln tc^ win> thus obtaining the conditional probability
            distribution of W^n j; ^ wln* The nu-*--*- hypothesis is
            rejected if W^n ^ W^n for the actual observed sample is
            too large according to this conditional distribution.
            4.5 Estimating the Covariance
            In Section 4.3, the asymptotic distribution of
            W£n |:T1 W^n for i = l,2,3 was established. In each case,
            depended on the underlying distributions F(*,*) and G(*)
            » i
            (the c.d.f. of (Xji» ^2i^ an<^ *^i ’ resPect i vely) . Hence, we
            can not perform a large sample test based on W in tí1 ?in
            unless i=l,2,3 is known.
            This section will discuss estimation for the components
            of j: i=l,2,3 which depend on the underlying distribution.
            These estimators, when substituted into the appropriate
            quantities they are estimating, provide asymptotically
            distribution-free statistics that can be used in the

            124
            hypothesis testing situations considered in this
            dissertation.
            For the variance-covariance matrix j; ^ and, thus,
            since it is identical, the term in j; ^ which depends on the
            underlying distribution is o ^ ^ = 12K^L^P* - K^L 2(3 / 4)^2
            (page 109). The dependence due to P , which was defined as
            P = Pr { ( j X2 1 ~y J — I X^ j — y J + I ^2 2-P I ~ | x ^ 2 ~ y | ^ 0 »
            (x^~ X21+ x 13" X23 ^ ^ } ~ M4 »
            was a result of the asymptotic covariance between TE* and
            T . Lemma 4.5.1 defines a consistent estimator for the
            nl
            quantity 12P (the asymptotic covariance of TEn and Tn ).
            First, though, we describe some notation which will be
            needed . Define
            ?i = (xii»x2i)»
            h(1)(Xi) = ¥(Ix2i— MI~Ixii-M| )»
            h(2)(Xi’Xj> = '•'t lX2i~MHXli~Ml + lX2j"MHXlj'Ml »
            h(3)(Xi) = f(xu- X2i) ,
            h(4)(Xi,xj) = ^xli* x2i+ Xlj- x2j>.
            h(1’3)(X.) = h(1)(Xi)h(3)(Xi),

            125
            h(1’4)(Xi,Xj) = h(1)(Xi)h(4)(Xi,Xj)
            + h(1)(X.j)h(4)(Xi,X.j ) ,
            h(2’3)(Xi,Xj) = h(2)(X.,Xj)h(3)(Xi)
            + h^2^(Xi,Xj)h^3^(Xj),
            h22,4)(?i’?i> =
            h(2)(Xi,Xj)h(4)(Xi,Xj)
            and
            h[2*4)(Xi,Xj,Xk) = h(2)(X.,Xj)h(4)(Xi,Xk)
            + h(2)(xi,xk)h(4)(x.,xj)
            + h(2)(Xi,Xj)h^4)(Xj »Xk)
            + h(2)(Xj,xk)h(4)(xi,xj)
            + h(2)(Xi,Xk)h(4)(Xj,Xk)
            + h(2)(Xj,xk)h(4)(xi,xk).
            The quantities h^2^(*,*), h^3^(») and h^4^(»,») are
            actually the kernels of U-statistics or kernels of
            U-statistics with an estimated parameter which are used in
            the representation of T and TE„ . (See page 74 for the
            nl nl
            exact U-statistic representation of Tn . A similar
            representation for TEn can be defined.) The quantities
            h^3,3)(*,»), h^’4^(»,»), etc., are needed to calculate to
            covariance between the kernels of the U-statistic
            representations of T and TE . A consistent estimator for
            111 nl
            Cov(T ,TE ), which will be defined in Lemma 4.5.1, can be
            nl nl
            viewed as estimating the exact covariance between TE'
            ll
            Tn^ using the sample covariance. In the proof of Lemma
            and

            126
            4.5.1, it will be argued that this estimator is a consistent
            estimator for the asymptotic covariance.
            Lemma 4.5.1: Under Hq ,
            CovaB^.T^) =
            4 "(1,3) , 4 "(1,4)
            “ 5 (n,-l) S
            (n^l) 1
            1
            , 4 "(2,3)
            ( n ! “1 ) S
            4U1 2) "(2,4) 2 "(2,4)
            ( n , -1 ) + (ti1-l ) 52
            1
            is a consistent estimator for 12P (the asymptotic
            covariance of TE„ and T ) where
            nl nl
            -0,3), _J_ ¡ hU,3)( , . -
            1 H 1 i = l '1
            (Dh(3)
            :(i ,4)_ i
            l l h
            2(nl) i 2
            ( 2 , 3 ) _ 1_
            l l h
            1 2 ( n 1) i (1,4)(X. ,X.) - h(1)h(4)
            (2 ,3)(X± ,Xj) - h(2)h(3)
            £( 2,4 ) _ 1
            1 6 ( 111) i 3
            l l l h(2,4)(X.,X.,X) - h(2)h(4)
            nn i ^ ( 2 , 4 ) = _J_
            2
            — y y h(2,4)(x.,x.) -
            n0 i h(2)h(4)
            h(1)= -L- l h(1)(X,)
            nl i=l
            (i.e., the actual sample

            127
            value of a U-statistic with kernel and analogous
            definitions for h^^ and h^^.
            Proof:
            Note that
            1 1
            Thus, the proof will be complete if it can be argued that
            This f o Hows
            directly since for a U-statistic, Un, based on a kernel, h ,
            •k p *
            + h by Hoeffding's Theorem (Hoeffding, 1961)
            â–¡
            Note that this is just one of many possible consistent
            estimators for 12P*. This estimator is presented because it
            worked well in the Monte Carlo study presented in Chapter
            Five. Although other estimators may appear to be
            reasonable, all too often in practice their determinant will
            be less than or equal to zero. Now we consider estimators
            for the va r i a nee-co va r i a nee matrix for W^, the quadratic
            form using CD.
            In looking at the variance-covariance matrix derived
            in Section 4.3, we notice immediately that the estimation
            needed here is more complicated than that of |^ . First, a
            consistent estimate for the asymptotic variance of CD (i.e.,
            4y) is needed. Secondly, we need to estimate two asymptotic
            covariances which are used in the calculation of o-j * y

            128
            ( i . e . ,
            The first estimation problem has already been taken care of
            in Section 2.4. The method of solving the second estimation
            k
            problem is similar to that used for estimating 12P . That
            is, the exact covariance between TE„ „ and CD is derived
            n 1 >nc
            and the sample quantities are then used in its
            calculation. Lemma 4.5.2 will present this estimator and
            argue that this is a consistent estimator for the asymptotic
            covariance. This estimator will be actually in the form of
            two estimators; one which is estimating the covariance of
            TE and CD and the other which is estimating the covariance
            nl
            of TEn and CD. This is equivalent to estimating the
            c
            quantities
            and
            C o v [ ( 1 - 2Yj), aijbij | 6jG(2,3)]
            First though, we describe some notation which will be
            needed. Let h^3^(X^) and h^^^(X^, X^ ) be defined as
            before. In addition, define
            h<5)(?j) *
            = h(3)(Xi)h(lc)(Xi,Xj)
            + h(3)(Xj)h(lc)(Xi,X:j) ,

            129
            h1<4'1C)Ui.?j.?k)
            = h(4)(Xi,Xj)h(lc)(Xi,Xk)
            + h(4)(Xi,Xj)h(lc)(Xj,Xk)
            + h(4)(Xi,Xk)h(lc)(Xi,XJ)
            + h(4)(Xi,Xk)h(lc)(Xj,Xk)
            + h(4)(xj,xk)h(lc)(xi,xk)
            + h(4)(Xj,xk)h(lc)(xi,xj),
            h2 ( 4 ’ 1 c } ( X± , Xj )
            = h(4)(X± , X,
            )h( lc) (X± ,Xj ) ,
            h(3>3c)(Xi,Xj) = h(3)(X1)hi3c)(Xi,Xj),
            h(4>3c)(Xi,Xj,Xk)
            = h(4)(xi,x.)h(3c)(x.,xk)
            + h(4)(Xi,Xj)h(3c)(Xj,Xk),
            h(5,2c)(xi>xj) = h(5)(xi)h(2c)(xi,x:j)
            + h(5)(Xj)h(2c)(Xi,X:j) ,
            and
            h(5,3c)(Xi ,Xj ) = h(5)(Xj)h(3c)(Xi,X:j) .
            Note, the quantities h^3c^(*,*)> h^2c3(*,*) and h^3c^(*,*)
            are actually the kernels in the U-statistic representation
            of CD given on page 113. The quantities h^3’3c^(• , • ) ,
            311 ^ 4 ’ 1 c )(•,*,•), e t c . , are needed to calculate the covariance
            between the kernels of the U-statistic representation of
            TEn and CD, and TEn and CD. In Lemma 4.5.2, the
            consistent estimator is now defined.

            130
            Lemma 4.5.2: Under HQ, a consistent estimator for
            V4y) '2 (3Xl)Vz [4Cov{'F(Xii-X21+xij-X2j), aijbij|6j-6j-l}]
            + K2(4y) ^ (X 2 ) ^ [2Cov{l-2y ,a b |6 e(2,3)}]
            is the following
            A -l/0 * Vor 4 A(3 1c') ^ D 1
            v (iy ) '2(3x y 2 [ ^ o , J. c ,/
            Y; [(n-l) ^ (n-
            4 ( n -2 ) a
            (4,1c)
            1 )
            + 2, - ^4»lc> +
            (n-1) ^2
            -1/ a y 2(n„- 1) A
            4nc *(3,3c) 4nc p(4,3 c)
            (n-1)(n j-1) (n-1) ^1
            + K2(4r) 2 (X2)'2
            (5,2c)
            2n, a,
            . 1 *(5,3c)
            (n-1) 4 j
            where
            4y is a consistent estimate for 4y,
            X i —
            n,
            1 n. + n ’
            1 c
            x2 -
            n, + n *
            1 c
            and 4
            pO.lc), “(4,1c) p ( 4,1 c )
            1 ’ ’1 ’ *2
            are summarized in the following table
            , etc. are U-statistics which

            131
            Table 4.1 Summarizing the U-statistics Used in Estimating
            the Covariance for
            U-Statistic Kernal Conditions on the <5' s
            2(3,10
            ^ i
            5i=6j=!
            ~(4,lc)
            g 1
            6i=6j=6k = 1
            2(4,lc)
            2
            h2(4-lc)(-,-)
            6i=6j=l
            2(3,3c)
            6i = l,6j e(2,3)
            2(4,3c)
            ^ 1
            h<4-3<=>(.,.,.)
            5 ± = 6j =1,6ke(2
            2(5,2c)
            h(5-2c)(-,.)
            (6i,5j)e(2,3)
            2(5,3c)
            g 1
            h(5-3c)(.,.)
            5i = l , 6 j e(2,3)
            Proof :
            Let the estimator defined in Lemma 4.5.2 be denoted by
            Cov(TE ,CD ). Notice that
            nl » c
            , * *
            |Cov(TE , C D
            nl*nc
            ) - ( 3 X j )V2
            lh r4(nl 2) ?(4,lc)
            , 4nc A(4,3c)
            (n-1) S
            .-ty? r2(nc 1') A ( 5,2 c ) ^ 2nl A ( 5 , 3 c )
            - 2 (V2 l-rbrr *
            + S
            ]}
            -+ 0.
            Thus, if we can show that

            132
            4 ( n ^ - 2 )
            ( n-1 )
            *(4 , lc )
            1
            4nc p(4,3 c)
            (n-1) i
            is a consistent estimator for
            4Cov(f(Xu- X21+ Xj.- X2J), alkblk|5i-6.-l},
            and that
            2(nc _1) A ( 5 , 2 c ) 2nl ^(5,3c)
            (n-1) + (n-1)
            is a consistent estimator for
            2Cov(l-2y., a..b..|ó.e(2,3)},
            1 J ij ij 1 ] J
            the proof will be complete. Recalling that
            nl p nc P
            — —> X^ and —-v X^, we observe that asymptotically,
            the coefficient in front of each £ terra is the probability
            necessary to uncondition the term so that within each
            estimator the terras can be combined appropriately. (See
            page 118 for a similar argument.) Also, since each
            estimator is a U-statistic, it follows that each is
            consistently estimating the appropriate covariance term and
            â–¡
            the proof of Lemma 4.5.2 follows

            1
            CHAPTER FIVE
            MONTE CARLO RESULTS AND CONCLUSION
            5.1 Introduction
            The first three chapters of this dissertation have been
            devoted to developing tests statistics for the purpose of
            testing for scale differences in censored matched pairs.
            Chapter Four used the statistics proposed in Chapters Two
            and Three to develop a vector of statistics designed to test
            for the more general alternative of location and/or scale
            differences. This chapter will investigate the performance
            of some of the test statistics proposed.
            In Section 5.2, a simulation study will be presented to
            compare the intermediate sample size performance of some
            members of the proposed class of statistics presented in
            Chapter Three and the CD statistic of Chapter Two.
            Similarly, Section 5.3 will present a simulation study for
            selected Wn vectors and a test statistic proposed by Seigel
            and Podger (1982). In each of these simulation studies, the
            asymptotic distribution of each test statistic is being
            used.
            133

            134
            5.2 Monte Carlo for the Scale Test
            In this section, nine statistics will be investigated
            to compare their performance under the null and alternative
            hypotheses. Three of the nine, are versions of the CD
            statistic, where each version uses a different estimator for
            the variance of CD (Section 2.4). The next three statistics
            are members of the class of statistics proposed in Section
            3.2, where the common location parameter was known, while
            the following three are members of the class of statistics
            proposed in Section 3.3, where the common location parameter
            was unknown and thus estimated. Table 5.1 summarizes the
            nine statistics considered in this Monte Carlo.
            In this Monte Carlo three bivariate distributions (each
            with common location (0,0)) were considered for generating
            /
            the bivariate samples. Although, generally the common
            location is not (0,0), it was used without loss of
            generality. The first distribution, the bivariate normal,
            was generated using the subroutine GGNSM of the
            International Mathematical and Statistical Library (IMSL).
            It allows specification of the variance-covarianee structure
            for the bivariate pairs. The remaining two distributions
            were generated using a technique for generating elliptically
            symmetric distributions proposed in Johnson and Ramberg
            (1977). The two distributions are both Pearson Type VII
            multivariate distributions which were generated in the
            following manner. For a sample of size n, 2n uniform [0,1]

            135
            Table 5.1 Summary of the Test Statistics Considered in the
            Monte Carlo for Scale.*
            Test
            Statistic
            1
            2
            3
            4
            5
            6
            Description
            CD statistic using Var^(CD)
            CD statistic using Va^CCD)
            CD statistic using Var^CCD)
            nl * n<
            = T_ + T
            T „ = 2 T + T
            Hi > n n! n.
            n 1 »n(
            = T + 2 T
            nl nc
            7
            TM_
            nl
            II
            O
            c
            T
            nl
            (M)
            + T^c
            00
            TM
            nl
            ’nc
            2T
            n
            (M)
            1
            + Tn
            9
            TM_
            nl
            ’nc =
            T
            nl
            (M)
            + 2Tn
            (i . e . ,L ^ L2 1)
            (i.e . , L ^ =2 , L2 = 1)
            (i.e. , L L = 1,L2 = 2)
            (i.e. , L ^ = 1 , L 2-1)
            (i.e..Lj-2,L2=1)
            (i.e . .Lj-l,L2 = 2)
            Section
            in Thesis
            2.4
            2.4
            2.4
            3.2
            3.2
            3.2
            3 . 3
            3.3
            3.3
            Tn,(M) denotes the T statistic of Section 3.3 which
            1 n j
            uses an estimate for the common location parameter.
            random variables (denoted IK i = 1,2 , . . . ,2n) were first
            generated and then the following transformations were
            applied:
            X1 * = (U2i.11/(1-V) ~ 1 ) 2 cos ( 2ttU2í )
            X2i = (u2i-l1/(1~V) “ 1)/2 sin(2TrU2i)
            where the parameter v (for bivariate pairs, v > 1) specifies
            a particular Pearson Type VII distribution. To generate
            bivariate pairs with scale parameters and 02 and
            correlation p, the following transformation was applied:

            136
            Xíi = °iXli
            and
            ^ 2 P a2Xl i n ^ ( f P ) ^
            2 V2 *
            2 i *
            The values of v which were chosen were v = 1.5 (which
            corresponds to a bivariate Cauchy distribution) and v = 3 (a
            distribution with moments and moderate tailweight).
            To generate the censoring random variable, the natural
            logarithm of the Uniform [0,B] distribution was used. The
            choice of B was made separately for each distribution with
            specific correlation p, so that under Hq approximately 25%
            of the total sample was censored in some manner. Three
            values for p were chosen: (1) p = .2 (weak correlation),
            (2) p = .5 (moderate correlation) and p = .8 (strong
            correlation). Note that the value of p affects the type of
            censoring occurring in the samples, that is, when p = .8,
            type 4 pairs dominate the observations which are censored,
            while when p = .2, type 2 and 3 pairs dominate the
            observations which are censored. Since the results
            presented in this section apply to the pattern of censoring
            described above, any conclusions drawn only apply to this
            form of censoring.
            In each case the null hypothesis was HQ: = a 2 and
            the alternative was Ha: a 2 > o^. The tests were conducted
            at the .05 level of significance using the asymptotic
            distribution for each test statistic. The first Monte Carlo
            study consisted of generating 1000 independent censored

            137
            samples of size 25, while the second utilized 1000
            2
            independent samples of size 40. In each, the value of
            2
            was 1.0 while the value of 02 was 1.0 (under HQ) or 2.0 or
            3.0 (each value corresponding to a different run of the
            fortran program listed in Appendix 2). Tables 5.2-5.4 give
            the results of the Monte Carlo for each distribution type
            with entries corresponding to the number of times, a
            statistic rejected HQ. The nine test statistics are
            numbered in accordance with the listing in Table 5.1. The
            standard deviation associated with each entry e can be
            estimated by ( e ( 1 0 0 0-e ) / 1 000 )^2.
            Inspecting Tables 5.2-5.4, we see that as the
            correlation increases between the components in the
            bivariate pairs, that the power increases for all the tests,
            regardless of the distribution considered. This exhibits
            the fact that the tests were designed to use the intra-pair
            information or at least some of it, in the case of the
            distribution-free statistics which correspond to columns
            4-9. The CD statistics (columns 1-3) which use more
            intra-pair and inter-pair information than the distribution-
            free statistics are performing the best across all of the
            distributions considered.
            Recall, the only difference in the CD statistics
            (columns 1-3) is the method of estimating the variance. The
            CD statistic corresponding to column 2 which uses Va^iCD)
            (an estimate for the asymptotic variance (4y) which is the
            variance of a conditional expectation, page 41 in this

            138
            Table 5.2 Approximate Powers of the Tests for the Bivariate
            Normal Distribution
            P = -2
            n
            ¿.
            °2
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            41
            49
            42
            45
            45
            44
            40
            40
            42
            25
            2.0
            312
            372
            337
            294
            268
            307
            287
            258
            292
            25
            3.0
            561
            619
            584
            526
            489
            553
            506
            471
            541
            40
            1.0
            35
            44
            34
            41
            39
            42
            45
            44
            41
            40
            2.0
            489
            523
            492
            412
            369
            445
            400
            361
            443
            40
            3.0
            790
            809
            795
            704
            665
            732
            699
            644
            731
            n
            2
            °2
            P
            = .5
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            46
            55
            45
            51
            55
            47
            53
            54
            48
            25
            2.0
            371
            429
            388
            342
            325
            358
            331
            309
            344
            25
            3.0
            630
            688
            657
            608
            568
            642
            590
            543
            615
            40
            1.0
            42
            47
            42
            47
            46
            47
            49
            51
            47
            40
            2.0
            538
            575
            547
            476
            445
            516
            475
            443
            517
            40
            3.0
            869
            885
            871
            780
            736
            835
            765
            721
            816
            n
            2
            °2
            P
            = .8
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            56
            61
            47
            53
            56
            53
            55
            54
            53
            25
            2.0
            546
            597
            549
            507
            482
            529
            495
            477
            528
            25
            3.0
            831
            877
            855
            796
            773
            827
            774
            747
            799
            40
            1.0
            50
            56
            46
            45
            48
            46
            50
            52
            48
            40
            2.0
            770
            795
            772
            67 1
            655
            719
            67 1
            6 4 6
            711
            40
            3.0
            974
            979
            977
            931
            921
            947
            914
            893
            935

            139
            Table 5.3 Approximate Powers of the Tests for the Bivariate
            Cauchy Distribution (Pearson Type VII, v=1.5)
            p = .2
            n
            ° 2
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            48
            64
            57
            50
            46
            51
            56
            57
            54
            25
            2.0
            222
            294
            269
            273
            269
            275
            252
            244
            262
            25
            3.0
            422
            521
            491
            468
            451
            460
            441
            431
            444
            40
            1.0
            38
            50
            4 6
            49
            48
            48
            54
            52
            51
            40
            2.0
            353
            399
            383
            365
            355
            362
            348
            338
            350
            40
            3.0
            632
            684
            665
            637
            611
            639
            613
            589
            623
            n
            2
            °2
            P
            = . 5
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            46
            67
            58
            52
            51
            56
            56
            46
            58
            25
            2 .0
            272
            343
            324
            307
            296
            327
            296
            281
            3 12
            25
            3.0
            495
            587
            57 1
            567
            537
            564
            516
            503
            532
            40
            1.0
            40
            51
            48
            45
            47
            42
            38
            41
            39
            40
            2.0
            424
            472
            458
            451
            424
            453
            410
            401
            430
            40
            3.0
            735
            773
            764
            741
            721
            757
            718
            682
            724
            n
            2
            °2
            P
            = .8
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            42
            61
            49
            55
            53
            53
            55
            54
            57
            25
            2.0
            405
            492
            467
            450
            438
            467
            424
            420
            436
            25
            3.0
            679
            778
            751
            762
            735
            770
            718
            698
            739
            40
            1.0
            49
            63
            57
            55
            54
            60
            58
            59
            62
            40
            2.0
            612
            665
            6 4 4
            656
            642
            675
            620
            601
            640
            40
            3.0
            907
            925
            924
            926
            917
            933
            908
            895
            920

            140
            Table
            5.4
            2
            0 2
            Approximate
            Pearson Type
            Powers of
            VII, v=3
            the
            P =
            Tests
            .2
            for
            the
            Bivariate
            n
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            52
            69
            56
            55
            54
            59
            54
            54
            53
            25
            2.0
            262
            330
            288
            268
            255
            280
            263
            242
            276
            25
            3.0
            483
            562
            514
            492
            470
            513
            464
            444
            492
            40
            1.0
            41
            46
            43
            46
            46
            47
            53
            51
            51
            40
            o
            •
            CM
            420
            460
            428
            380
            372
            392
            364
            350
            382
            40
            3.0
            737
            770
            753
            672
            649
            711
            655
            623
            684
            n
            2
            a 2
            P
            = .5
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1.0
            48
            62
            52
            50
            52
            52
            59
            61
            58
            25
            2.0
            319
            380
            350
            321
            314
            330
            298
            290
            315
            25
            3.0
            567
            648
            618
            574
            554
            599
            552
            527
            570
            40
            1.0
            46
            54
            48
            48
            47
            49
            49
            53
            47
            40
            2.0
            497
            540
            506
            452
            437
            477
            446
            426
            461
            40
            3.0
            802
            833
            812
            767
            743
            790
            749
            719
            767
            n
            2
            °2
            P
            = .8
            1
            2
            3
            4
            5
            6
            7
            8
            9
            25
            1 .0
            49
            66
            51
            54
            53
            54
            54
            54
            54
            25
            2.0
            484
            555
            513
            504
            497
            516
            484
            464
            490
            25
            3.0
            782
            835
            816
            769
            760
            791
            754
            744
            775
            40
            1.0
            44
            56
            48
            53
            54
            54
            63
            64
            62
            40
            2.0
            713
            742
            725
            685
            674
            695
            649
            642
            663
            40
            3.0
            953
            964
            958
            941
            927
            947
            933
            920
            942

            1
            141
            thesis) has some problem maintaining the significance level
            under H and thus is not recommended. The CD statistics
            o
            corresponding to columns 1 and 3 are maintaining the .05
            level under HQ, with the statistic corresponding to column 3
            performing the best over the alternatives. Recall, the CD
            statistic which uses Var^(CD) (column 1) is using a
            U-statistic to estimate the asymptotic variance (4y), while
            the CD statistic which uses Var-j(CD) (column 3) is
            2 4(n-2)
            estimating the exact variance that is — rr a + —7 CT Y •
            ° n(n-l) n(n-l)
            For the CD statistics, only once was there a negative
            estimate for the variance which occurred for the Pearson
            2
            Type VII, v=3 distribution with n=25 and 0£ = 2.0 . One
            basic disadvantage of the CD statistics is the fact they
            require the use of a computer to perform the calculations
            for even moderate sample sizes. The CDSTAT subroutine of
            the fortran program listed in Appendix 2 could be used. If
            a computer is not available or the necessary knowledge to
            use it to program the calculations for the CD statistic,
            then a distribution-free test could be recommended.
            Of the distribution-free tests in columns 4-9, the
            tests which uses weights of 1^ = 1 and 1*2 = 2 (columns 6 and 9)
            appear to be performing the best. Column 6 corresponds to
            the statistic for the case when the common location
            parameter is known, while column 9 corresponds to the
            statistic for the case when the common location parameter is
            unknown. The corresponding test statistics using equal
            weights, that is, L^= 1*2= 1 (column 4 corresponds to the

            1
            142
            case of the known location parameter, while column 7
            corresponds to the case of the unknown location parameter)
            follow closely behind in terms of power. If the common
            location is known, in each case the power is improved by
            using that known value (column 6 for weights Lj = l and L,2 = 2
            or column 4 for equal weighting). When the correlation is
            high, note that all the statistics considered in this Monte
            Carlo are performing well.
            In summary, the best statistic to use is CD using
            Var^CCD) when the necessary computations which require a
            computer can be done. If it is not possible to calculate
            the CD statistic, the distribution-free test statistic using
            weights of L^= 1 and 1.2= 2 (corresponding to column 6) is
            recommended when the location parameter is known. If the
            location parameter is unknown, then the distribution-free
            test statistic using weights of L^= 1 and L,2 = 2
            (corresponding to column 9) is recommended.
            5.3 Monte Carlo for the Location/Scale Test
            In this section, twelve statistics will be investigated
            to compare their performance under the null and alternative
            hypotheses considered in Chapter Four. The alternative
            hypotheses studied here include alternatives for location
            differences only, for scale differences only and for
            location and scale differences. The first nine statistics

            143
            which will be considered are various versions of the
            quadratic forms presented in Chapter Four. The tenth
            statistic was introduced by Seigel and Podger (1982) and
            will be defined in this section. The last two statistics,
            which are from Chapters Two and Three, are included here to
            determine their performance under the alternatives
            considered in this section. The first is the distribution-
            free statistic TM„ „ which uses weights of L,= 1 and L0= 2
            nl>nc 1 2
            (corresponding to statistic 9 in Section 5.2). The second
            A
            is the CD statistic which uses Var^iCD) to estimate the
            variance of CD (corresponding to statistic 3 in Section
            5.2).
            Of the nine quadratic forms from Chapter Four, the
            first
            three are versions of
            ?2n
            I21 ~2n
            where W2n is
            t he
            vector
            of statistics which
            uses
            T E „
            nl >nc
            (which tests
            for
            location)'and TM„ „ (which tests for scale) with the
            nl ’ c
            common location estimate. The three different versions
            correspond to different choices for the weights, K^, K2n’
            Lfn and L2n, used in forming W2n* The next three quadratic
            - ? — i
            forms are versions of W^n W^n where is the vector of
            statistics which uses TE „ and T „ (which tests for
            nl’nc nl’nc
            scale) with the separate location estimates. Similar to the
            first three quadratic forms, these three different versions
            correspond to different choices for the weights K^n, K2n,
            Lfn and l^^ The choices of weights used here are identical
            to those used in W2n and will be defined shortly. The last
            three quadratic forms correspond to different versions of

            144
            W3n j-3* w3n w^ere W3n the vector of statistics which uses
            TE _ and CD* (the standardized CD statistic) with
            nl ’ nc
            Var^CCD) as an estimate of the variance of CD. Again, the
            three versions correspond to different choices for the
            weights Kln and K^n* For the weights Kln, K2n’ Lln and L2n’
            the following choices were used:
            (1) Kjn= K2n= ^ln= ^2n=^’ (This amounts to just summing
            the standardized statistics to form the
            location statistic and the scale statistic.)
            (2) Kj„- L
            1
            1 n"
            and K9 = L9 =
            2 n 2n n,+ n
            (This
            1 c 1 c
            weights each statistic proportionally to the
            and
            sample size used in its calculation and will be
            denoted as SS weights.)
            o (TE )
            n
            Kln a(TE
            1
            ni’nc
            , K
            a ( TE )
            n
            c
            2n" a(TE ) ’
            ni’nc
            L, =
            1 n
            o(T )
            nl
            o(T
            ni’nc
            -r and K„ =
            ) 2 n
            o(T )
            n
            o(T ~)
            nl’nc
            where a(TE ), a(TE ) and o(TE )
            n1 nc nl’nc
            represent the standard deviations of TE„ , TE„ and
            n 1 nc
            TEn + TEq , respectively, under the null
            1 c
            hypothesis. Similarly, o(Tn ), o(Tn ) and
            u 1 c
            a(TR n ) represent the standard deviations of

            145
            Tn , Tn and Tn n , respectively, under the null
            hypothesis. (This weights each statistic
            proportionally to its null standard deviation and
            will be denoted as STD weights.)
            *
            For the quadratic forms using CD (i.e., versions of
            W3n 1*3* W3n^ on^y tlie weights and K2n were used since
            the definition of CD did not include any weights. For the
            quadratic forms based on ^2n anc^ ^3n’ t'ie t hree choices for
            the weights, K^n, K^, L^n and L2n, correspond to the three
            versions of each statistic which will be presented. Table
            5.5 summarizes the twelve statistics considered in this
            Monte Carlo.
            The test which was proposed by Seigel and Podger (1982)
            can be viewed as a special case of the procedure commonly
            referred to as the log rank, method proposed by Mantel
            (1966). This test assumes the null hypothesis that the
            survival curve for is identical to that of X2 ^ at all
            points. The alternative for this test is that differences
            exist between the two curves. Note, that this is a more
            general alternative than the alternatives specified in
            Chapter Two and Three but the test was included to determine
            how well it performed for the alternatives considered
            here. Now, to define the test statistic. Let n represent
            the total number of type 1, 2, and 3 pairs and let nu be
            the number of pairs for which X^ > X2^. Similarly, define
            nt to be the number of pairs in which Xli < X2i
            Note that

            146
            Table 5.5 Summary of the Test Statistics Considered in
            Monte Carlo for Location and/or Scale
            Alternatives.
            Test
            Statistic
            Description
            Section
            in Thesis
            1
            ?2n
            W2n
            with
            Kln"K2n“Lln'L2n L
            4.2
            2
            ^ 2 n
            k1
            ~2n
            with
            SS weights
            4.2
            3
            ?2n
            Í;1
            W2n
            with
            STD weights
            4.2
            4
            Win
            ÍT1
            Win
            with
            K! ~=K9„=L, „=L 9 =1
            In 2n In 2n
            4.2
            5
            «in
            ir1
            Win
            with
            SS weights
            4.2
            6
            win
            i:1
            Win
            with
            STD weights
            4.2
            7
            ~3n
            h1
            W 3 n
            with
            Kln=K2n=1
            4.3
            8
            -3 n
            hl
            ~3 n
            with
            STD weights
            4.3
            9
            ~3n
            h1
            ~3n
            with
            SS weights
            4.3
            10
            Siegel
            Podger Statistic
            5.3
            11
            â„¢n
            1 ’nc
            " Tc
            (M)
            ll
            + 2T
            nc
            3.3
            12
            CD
            statistic
            using Varg(CD)
            2 .4
            nu + nt = n. The test proposed by Seigel and Podger (1982)
            is to compare the observed frequencies nu and n^ , against
            the expected values of n/2 (under HQ) using the binomial
            distribution or when appropriate, an approximate large
            sample distribution. A suggested statistic which would be
            appropriate for the approximation is McNemar’s statistic
            which could be defined here as
            TSP = ^nt " nu> ^n*
            Under HQ, Tgp has a limiting y/jx
            distribution.

            147
            For this Monte Carlo study, the uncensored bivariate
            samples were generated using the techniques described in
            Section 5.2. Those techniques generate bivariate pairs with
            scale parameters o^ and and correlation p. Without loss
            of generality, the location parameter for X2^ was chosen to
            be 0, while location parameter for X^ was u2• The
            censoring random variables were also generated in the same
            manner as in Section 5.2, so that under Hq approximately 25%
            of the total sample was censored in some manner. The values
            for p considered in this section are .2, .5 and .8 (as in
            Section 5.2).
            For
            each Monte
            Carlo run
            the
            null hypothesis was
            V
            p 2 and 0 j
            0 2 > and
            the
            alternative was
            V
            yl
            ^ y 2 and/or
            0 1 / 0 2 •
            The
            tests were conducted at
            t he
            .05
            level of significance
            using the asymptotic
            distribution for each test statistic plus consistent
            variance-covariance estimators defined in Section 4.5, where
            appropriate. This study consisted of generating 500
            independent censored samples of size 35. To produce the
            alternatives considered here, the value of 0^ was 1.0, while
            the value of 0^ was 1.0, 2.0, or 3.0. Similarly, the value
            of \¡ 2 was 0, while the value of was 0.5 or 1.0. Only
            seven alternatives (in addition to the null hypothesis),
            were chosen for this study and correspond to the following
            choices for y^ and 0^ :
            (1) y^ = 0.0 and 0^ = 2.0,
            (2) y ^ = 0.0 and =
            3.0,

            148
            (3)
            yi= 0.5
            and
            i -
            1.0,
            (4)
            y i= 1-0
            and
            °t ■
            1.0,
            (5)
            y i= 0.5
            and
            •! '
            2.0,
            (6)
            yi= 0.5
            and
            â– 
            3.0
            (7)
            O
            •
            »—H
            II
            »—H
            3.
            and
            II
            CM CM
            O
            3.0.
            Tables 5.6-5.8 give the results of the Monte Carlo for each
            distribution type with entries corresponding to the number
            of times a statistic rejected Hq . The twelve statistics are
            numbered in accordance with the listing in Table 5.5. The
            column headings Null, Scale, Location and Location/Scale
            refer to the type of bivariate pairs being generated and the
            type of alternative that it reflects. The standard
            deviation associated with each entry e can be estimated by
            fe(500 - e)/500)^2 . Covariance estimates between certain
            entries were also estimated but are not reported here. The
            following discussion refers only to statistics 1 to 10 in
            Tables 5.6-5.8. The discussion of statistics 11 and 12 will
            follow.
            Inspecting Tables 5.6-5.8, we see that as the
            correlation increases between the components in the
            bivariate pairs that the power increases for all the
            tests. Similar results were observed in the Monte Carlo in
            Section 5.2. For the null hypothesis (column 1), all the
            tests are maintaining the significance level under Hq fairly
            well, although, the levels for the tests using CD show more

            149
            Table 5.6 Number of Rejections in 500 Replications for the
            Bivariate Normal Distribution
            Null Scale Location Location/Sca1e
            Vx = 0.0
            0.0
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .0
            Stat
            a¡= 1.0
            2.0
            3.0
            1.0
            P=.2
            1 .0
            2.0
            3.0
            3.0
            1
            25
            1 10
            241
            220
            467
            258
            333
            451
            2
            33
            107
            223
            220
            461
            259
            330
            453
            3
            29
            97
            203
            153
            409
            231
            295
            397
            4
            30
            105
            235
            224
            471
            230
            293
            434
            5
            35
            108
            233
            230
            470
            239
            299
            437
            6
            40
            104
            216
            188
            436
            228
            288
            404
            7
            32
            157
            299
            231
            472
            268
            369
            459
            8
            34
            148
            294
            233
            4 6 6
            266
            370
            459
            9
            33
            140
            289
            179
            418
            221
            314
            400
            10
            24
            26
            29
            227
            465
            204
            189
            412
            11
            22
            177
            343
            21
            7
            126
            243
            60
            12
            18
            222
            379
            m
            •
            II
            Q.
            2
            125
            266
            128
            1
            25
            124
            302
            295
            494
            355
            409
            488
            2
            30
            123
            287
            300
            493
            352
            408
            486
            3
            31
            115
            256
            233
            477
            325
            391
            470
            4
            25
            127
            298
            290
            493
            322
            388
            481
            5
            36
            129
            278
            304
            496
            332
            383
            481
            6
            40
            122
            263
            278
            481
            315
            368
            472
            7
            36
            185
            361
            308
            493
            366
            427
            489
            8
            35
            181
            363
            311
            494
            364
            425
            487
            9
            32
            180
            355
            257
            481
            319
            400
            471
            10
            23
            25
            35
            295
            492
            278
            265
            469
            1 1
            28
            226
            384
            45
            19
            209
            327
            124
            12
            19
            268
            403
            5
            3
            156
            296
            149

            150
            Table 5.6 - continued.
            Null
            Scale
            Location
            Location/Scale
            u 1 = o.o
            0.0
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .0
            S tat
            o|= 1.0
            2.0
            3.0
            1 .0
            p = . 8
            1 .0
            2.0
            3.0
            3.0
            1
            28
            239
            405
            460
            500
            475
            487
            500
            2
            33
            243
            399
            4 6 6
            500
            477
            485
            500
            3
            30
            214
            375
            434
            500
            472
            483
            500
            4
            22
            23 1
            418
            455
            500
            463
            483
            500
            5
            37
            234
            394
            470
            500
            469
            482
            500
            6
            41
            215*
            368
            448
            500
            465
            482
            500
            7
            39
            307
            460
            461
            500
            478
            493
            500
            8
            41
            303
            459
            469
            500
            480
            492
            500
            9
            38
            307
            454
            439
            500
            469
            488
            500
            10
            31
            37
            45
            459
            500
            440
            414
            500
            11
            25
            322
            445
            74
            31
            347
            445
            198
            12
            22
            361
            477
            7
            2
            230
            413
            236
            * indicates one covariance matrix was not positive definite
            Table 5.7 Number of Rejections in 500 Replications for the
            Bivariate Cauchy Distribution
            Null Scale Location Location/Scale
            o
            •
            o
            II
            3.
            o
            •
            o
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .0
            Stat
            p2= 1.0
            2.0
            3.0
            1 .0
            P=.2
            1.0
            2.0
            3.0
            3.0
            1
            28
            92
            197
            98
            304
            167
            258
            354
            2
            31
            99
            204
            113
            317
            175
            260
            355
            3
            33
            95
            193
            107
            284
            170
            245
            327*
            4
            33
            96
            197
            97
            306
            167
            243
            337
            5
            36
            105
            206
            115
            333
            176
            250
            348
            6
            37
            90
            199
            130
            320
            177
            242
            330
            7
            42
            121
            244
            114
            328
            173
            269
            344
            8
            38
            123
            244
            120
            348
            179
            269
            351
            9
            37
            121
            245
            122
            331
            167
            254
            334
            10
            26
            27
            29
            156
            368
            141
            137
            320
            11
            35
            169
            298
            27
            18
            152
            266
            181
            12
            35
            198
            322
            26
            17
            159
            280
            212

            151
            Table 5.7 - continued.
            Null Scale Location Location/Scale
            P j= o.o
            o
            •
            o
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .
            S ta t
            a?- 1.0
            2.0
            3.0
            1 .0
            1 .0
            2.0
            3.0
            3 .
            P=.5
            1
            22
            106
            224
            140
            374
            222
            308
            421
            2
            22
            117
            233
            158
            390
            232
            306
            419
            3
            30
            110
            209
            141*
            360*
            219*
            305
            391
            4
            24
            112
            230
            142
            377
            203
            289
            397
            5
            30
            114
            235
            164
            402
            216
            286
            406
            6
            40
            110
            221
            176*
            417*
            216
            293
            394
            7
            32
            156
            280
            150
            385
            224
            309
            401
            8
            33
            152
            273
            171
            399
            235
            316
            406
            9
            35
            150
            269
            173
            412
            210
            309
            394
            10
            28
            30
            34
            196
            430
            204
            183
            400
            11
            24
            190
            347
            35
            27
            192
            323
            243
            12
            32
            215
            364
            25
            14
            165
            310
            230
            p = . 8
            1
            15
            167
            334
            246
            449
            367
            432
            484
            2
            20
            170
            338
            281
            471
            366
            430
            484
            3
            29*
            158
            k
            312
            270*
            459
            k k
            363
            429
            472
            4
            21
            165
            325
            250
            455
            355
            413
            479
            5
            27
            170
            342
            312
            476
            366
            418
            480
            6
            39
            159*
            322*
            315
            486
            361
            423
            477
            7
            45
            223
            394
            271
            464
            342
            420
            474
            8
            36
            228
            390
            328
            478
            363
            427
            480
            9
            34
            226
            389
            319
            483
            355
            424
            473
            10
            26
            31
            54
            361
            489
            368
            336
            487
            1 1
            29
            279
            435
            48
            55
            299
            435
            338
            12
            32
            297
            446
            27
            13
            243
            401
            307
            * indicates one covariance matrix was not positive definite
            ** indicates two covariance matrices were not positive
            definite

            152
            Table 5.8 Number of Rejections in 500 Replications for the
            Bivariate Pearson VII, v=3
            Null Scale Location Location/Scale
            O
            •
            o
            II
            3.
            0.0
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .0
            Stat
            a?- 1.0
            2.0
            3.0
            1 .0
            1 .0
            2.0
            3.0
            3.0
            P-.2
            1
            28
            99
            224
            220
            462
            263
            314
            445
            2
            32
            102
            223
            219
            461
            262
            313
            445
            3
            34
            102
            208
            168
            4 16
            223
            286
            405
            4
            29
            104
            222
            212
            472
            248
            285
            440
            5
            37
            103
            221
            223
            472
            257
            290
            439
            6
            39
            106
            217
            201
            453
            234
            277
            413
            7
            38
            139
            277
            227
            469
            275
            339
            452
            8
            38
            137
            273
            229
            468
            274
            334
            450
            9
            39
            134
            264
            199
            443
            221
            293
            409
            10
            30
            26
            29
            227
            469
            200
            193
            427
            11
            27
            179
            320
            26
            7
            144
            236
            69
            12
            32
            198
            355
            16
            4
            130
            253
            132
            P=.5
            1
            25
            112
            261
            296
            491
            338
            402
            488
            2
            26
            122
            253
            297
            491
            337
            399
            487
            3
            29
            115
            226
            230
            467
            297
            377
            471
            4
            25
            118
            249
            301
            495
            308
            378
            484
            5
            33
            123
            251
            314
            495
            313
            379
            486
            6
            35
            113
            229
            275
            484
            292
            361
            472
            7
            32
            173
            329
            309
            494
            341
            408
            487
            8
            33
            167
            324
            313
            495
            341
            407
            489
            9
            35
            166
            315
            276
            486
            297
            379
            4 6 4
            10
            28
            27
            27
            324
            495
            288
            270
            486
            11
            35
            208
            358
            38
            20
            205
            303
            120
            12
            28
            240
            391
            14
            5
            145
            278
            143

            153
            Table 5.8 - continued.
            Null
            Scale
            Location
            Location/Scale
            ul =
            o
            •
            o
            0.0
            0.0
            0.5
            1 .0
            0.5
            0.5
            1 .0
            S ta t =
            1.0
            2.0
            3.0
            1 .0
            p = . 8
            1 .0
            2.0
            3.0
            3.0
            1
            22
            201
            371
            447
            500
            476
            486
            499
            2
            24
            203
            368
            455
            500
            475
            485
            499
            3
            30
            190
            335*
            414
            500
            461
            477
            496
            4
            17
            182
            368
            445
            500
            461
            478
            499
            5
            28
            189
            364
            467
            500
            467
            479
            499
            6
            33
            185*
            329
            446
            500
            463
            476
            497
            7
            38
            267
            432
            450
            500
            4 66
            435
            499
            8
            34
            266
            430
            463
            500
            475
            486
            499
            9
            37
            257
            425
            442
            500
            458
            474
            497
            10
            25
            28
            41
            463
            500
            454
            422
            499
            11
            34
            305
            441
            58
            31
            309
            421
            189
            12
            38
            325
            467
            17
            7
            217
            384
            222
            indicates
            one
            covariance matrix
            was not
            positive definite
            fluctuation than the others. Of the CD tests, the
            fluctuation of the null power is greatest for the statistic
            in row 7 where it varies from 32 rejects (for the normal
            distribution with p=.2) all the way to 45 rejects (for the
            cauchy distribution with p=.8).
            For the alternatives in which the bivariate pairs were
            generated with scale differences only (i.e., u^= 0*0 and
            = 2.0), or p^= 0.0 and = 3.0), the first obvious
            conclusion is that the Seigel Podger statistic (row 10) is
            definitely not appropriate because it tests for location and
            not scale differences. As one would expect from the results
            jlf
            in Section 5.2, the quadratic forms using CD as a scale

            154
            statistic (rows 7-9) are performing better than the
            quadratic forms using the distribution-free scale statistic
            TM (rows 1-3) or its analog which uses the separate
            nl’ c
            location estimates (rows 4-6). No basic differences exist
            between the three quadratic forms which use CD for these two
            alternatives .
            For the alternatives in which the bivariate pairs were
            generated with location differences only (p ^ = 0*5 and
            a^= 1.0, or 1.0 and a|= 1.0), in general, the Seigel
            Podger statistic (row 10) is performing the best because it
            specializes in this type of alternatives. If we look at the
            bivariate Pearson VII distribution with v=3 and the
            bivariate normal distribution only, the statistics
            corresponding to rows 1, 2, 4, 5, 7 and 8 are all performing
            equivalently to the Seigel Podger statistic. It is in the
            bivariate cauchy distribution where the Seigel Podger
            statistic seems to have a slight advantage. Note that the
            statistics corresponding to rows 1, 4 and 7 use the equal
            weighting scheme while the statistics corresponding to rows
            2, 5 and 8 use the sample size weighting scheme in forming
            the corresponding Wn vector. A possible reason why the
            standard deviation weighting scheme seems to diminish a
            quadratic form's performance is that the variance for the
            term TEn is relatively small compared to the variance of
            TE
            _ . Thus, in the linear combination K, TE„ + K0 TE
            n ^ lnn^znn(
            most of the weight is being given to TEn . Popovich (1983

            155
            ) observed in his Monte Carlo, that this, in some cases,
            reduces the performance of the location statistic.
            Now we consider the alternatives in which the bivariate
            pairs were generated with location and scale differences
            (^2= 0.5 and = 2.0, or p^ = 0.5 and = 3.0, or p^= 1.0
            and = 3.0). For these alternatives, the statistics
            corresponding to rows 7 and 8 are performing the best
            overall. The statistics corresponding to rows 1 and 2 are
            performing equivalently to rows 7 and 8, except for the
            alternative where y^ = 0.5 and = 3.0 for the bivariate
            Pearson VII distribution with v=3 and p=.2, and the
            bivariate normal distribution with p-.2 or p=.5 . This
            possibly is reflecting the fact that CD performs slightly
            better than the distribution-free scale statistic TM
            nl >nc
            when scale differences exist and, that, for this alternative
            (i.e., y^= 0.5 and = 3.0) large scale differences
            exist. As the correlation increases within any
            distribution, we see that all the statistics are performing
            moderately well and that for the last alternative where
            Ui- 1.0 and = 3.0 when p=.5 or p=.8 no differences exist
            between the ten statistics in general.
            In summary, the recommended statistic is the quadratic
            form which uses CD and the sample size (SS) weights for
            TEn^ n (row 8). This statistic provides the best power in
            general for the alternatives considered here. The statistic
            corresponding to the quadratic form which uses CD and equal
            weights for TEn n (row 7) performs for the most part
            1 ’ c

            156
            equivalently, but is not recommended due to its higher
            fluctuation of levels under the null hypothesis. Note, if
            instead of the more general alternative hypothesis
            considered here (i.e., Ha: ^ p£ and/or ^ 02)» we can
            restrict the alternative to be more specific (i.e.,
            Ha: 02 # c^) then, in many cases, the power of the test can
            be improved by using a statistic which is specifically
            designed for that alternative.
            Finally, we turn our attention to the last two
            statistics (11 and 12) included in this Monte Carlo. These
            statistics were included for two reasons. The first reason
            was to investigate their performance when the bivariate
            pairs were generated with equal marginal scale parameters
            but unequal marginal locations. The second reason was to
            determine what effect unequal marginal locations had on the
            power of the tests when the bivariate pairs had unequal
            scale. Looking at the columns labeled Location, we see that
            both tests are fairly robust when the components of the
            bivariate pairs have equal scale but unequal marginal
            locations. From the last three columns of each table, we
            observe that slight differences in the locations parameters
            do not affect the power of the scale statistics appreciably
            but as the location differences become more pronounced the
            power of each test is dramatically reduced. In conclusion,
            if slight differences exist between the location parameters

            157
            for the marginal distributions, the tests for scale still
            perform appropriately but as the differences increase the
            tests have definite drawbacks

            APPENDIX 1
            TABLES OF CRITICAL VALUES FOR TESTING FOR DIFFERENCES
            IN SCALE
            The tables in this appendix list the critical values of
            T = T + T
            nl’nc nl nc
            (i.e., T with L, = L~ = 1) for .01, .025, .05 and .10
            n1’nc 12
            levels of significance for n^ = 1,2,..., 1 5 and
            n2 = 1,2,3,. . . 1 0 . These tables are also appropriate for
            TM _ , since T
            n 1 ’ nc
            nl ’ nc
            = TM
            nl ’ nc
            or larger values of n^ or
            nc, the asymptotic normal distribution of Tn n (and
            TMn n ) could be used. When n^= 0 or nc= 0, the critical
            values can be obtained from the critical values for the
            Wilcoxon signed rank distribution based on n^ or nc
            observations (respectively).
            The critical values for this test statistic were
            derived for each n^ and nc by convoluting two Wilcoxon
            signed rank statistics (based on n^ observations and n£
            observations). Thus, it follows that the critical values
            for a test based on n,= a and n = b are the same as the
            1 c
            critical values for a test based on n,= b and n = a
            1 c
            observations. Therefore, the tables can also be viewed as
            listing the critical values for n^ = 1,2, . . . , 1 0 and
            nc = 1,2 , ...,15. These
            test H : a,=o0 versus H
            o 1 2 a
            values are tabled for the
            a^<02* To determine the
            critical values for the alternatives H : Oj>02 or
            Ha: o^o 2, the symmetry of T about
            1 » nr
            158

            159
            n,(n.+ 1) + n (n +1)
            1 1 c c
            4
            could be used to calculate the
            necessary cutoffs.
            Since the Wilcoxon signed rank distribution is
            discrete, exact .01, .025, .05 and .10 level critical values
            do not always exist. These tables list the following:
            1) The critical value (c) for a specific a level,
            such that P(Tn n >c} 2) The attained significance level (p-value) of
            each critical value is given in the parentheses
            (i.e., P{T >c} = p-value).
            nl»nc
            3) The attained significance level of the next
            closest critical value is given in the square
            brackets (i.e., P{T >(c-l)}).
            nl ’nc
            For example, let n^= 10 and n£= 5, the critical value
            for a .05 level test would be 53. The attained signficance
            level for the test would actually be .048. The next closest
            critical value would be 53-1 = 52, with an attained
            signficance level of .059.
            When n^ and nc are both very small, (generally less
            than 3), many times a critical v^lue does not exist for a
            specific level of significance. Then, the value in the

            160
            bracket is P(T „ >ra) where m
            nl ’ nc
            n (n +1 ) + n (n +1 )
            1 1 c c
            2
            the largest value
            could
            be ) .

            161
            a
            n 1 = 1
            n 1 =2
            nl = 3
            01
            —-( ) [ • 250]
            — ( ) [ .12 5]
            — ( ) [ .063]
            025
            — ( ) [ . 250]
            — ( ) [ • 1 2 5 ]
            — ( )[. 06 3]
            05
            —-( ) [ .250]
            —-( )[ .125]
            — ( ) [ .063]
            10
            —( >[.250]
            nl=4
            ---( ) [ • 1 25 ]
            nl=5
            7 (.063)[.183]
            rij =6
            01
            — ( ) [ . 031]
            — ( ) [ .016]
            22 (.008)[.023]
            025
            — ( ) [ .031]
            16 ( .016) [ .047]
            21 ( .023 ) [ .039]
            05
            11 ( .031 ) [ .094]
            15 (.047)[.078]
            20 (.039)1.063]
            10
            10 (.094)1.156]
            nl=7
            14 ( .078 )[ . 125]
            n i =8
            18 ( .094 ) [ .133]
            nl = 9
            01
            29 ( .004 ) [ .012]
            35 ( .010) [ .016]
            43 ( .008) [ .012]
            025
            27 ( .020) [ .031]
            33 ( .023)[.033]
            40 (.023)[.032]
            05
            25 (,047)[.066]
            31 ( .047) [ .064]
            38 (.043)1.057]
            10
            23 ( .094 ) [.129]
            n ^ = 1 0
            29 ( . 086 ) [.111]
            = 1 1
            35 ( . 092 ) [.113]
            n j = 1 2
            01
            51 ( .008) [.012]
            60 ( . 008)[.011]
            69 ( .009)[.012]
            025
            48 (.02 1 ) [ .028 ]
            56 (.024)[.030]
            65 (.024)1.029]
            05
            45 (.047)[.059]
            53 ( . 046)[ . 056]
            '62 (.042)[.051]
            10
            42 (.088)[ .106]
            n j = 1 3
            49 ( .095 ) [.111]
            n j = 1 4
            57 ( .095 ) [ .109]
            n L = 15
            01
            79 (.009) [.012]
            90 ( .009 ) [.Oil]
            101 ( .010) [ .012]
            025
            75 ( .022) [ .026]
            85 (.023)1.027]
            96 (.022)1.026]
            05
            71 ( .044) [ .051]
            80 (.049)[.056]
            91 (.044)1.050]
            10
            66 (.090)1.101]
            75 (. 092 H.103]
            84 (.099)[.109]

            162
            n
            c
            = 2
            a
            nl=2
            ni=3
            II
            r—t
            e
            01
            —-( )[.063]
            —
            ( ) [ .031]
            — -( ) [ .016]
            025
            — ( ) t • 0 6 3 ]
            —
            ( ) [ • 0 3 1 ]
            13 ( .016)[. 047]
            05
            — ( ) [ .063]
            9
            ( .031 ) [ .094 ]
            12 (.047)[.094]
            10
            6 ( .063 )[.188]
            n 1 = 5
            8
            ( .094 ) [ . 188]
            n ^ = 6
            11 (.094)[.172]
            nl = 7
            01
            18 ( .008)[.023]
            24
            ( .004 ) [.012]
            30 ( . 006)[.012]
            025
            17 ( .023) [ .047]
            22
            ( .023 ) [ .043 ]
            28 ( .021 ) [ .033 ]
            05
            16 (.047)[.086]
            21
            (.043)[.066]
            26 (.049)[.070]
            10
            15 ( . 08 6 ) [ .133]
            n j =8
            19
            ( .098) [ . 141 ]
            nl=9
            24 ( .098 ) [ . 131 ]
            rij-10
            01
            37 ( .006 ) [.Oil]
            44
            ( .008) [ .012]
            52 ( .009)[ .012]
            025
            34 ( .024) [.035]
            41
            (.024) [.033]
            49 (. 022 )[. 029 ]
            05
            32 (.049) [ .066]
            39
            ( .044 ) [ . 058]
            46 ( .048) [ .060]
            10
            30 ( . 088 )[.113]
            nj-11
            36
            ( . 093 ) [ . 115]
            = 1 2
            43 (. 090)[. 108]
            = 1 3
            01
            61 (.008)[ .011]
            70
            ( .010)[.012]
            80 ( .010) [ .012]
            025
            57 (.024)[.031]
            66
            ( .024 ) [ .029 ]
            76 (.022)[.027]
            05
            54 (.047)[.057]
            63
            ( .043 ) [ .051 ]
            72 (.044)[.05 1 ]
            10
            50 ( . 100 ) [ . 112]
            n i = 14
            58
            ( . 096 ) [ .110]
            iij-15
            67 ( .090) [ .102]
            01
            91 ( .009)[.011]
            103(. 008)[ .010]
            025
            86 (.023)1.027]
            97
            ( .022 ) [ .026 ]
            05
            81 ( .049) [.056]
            92
            ( . 045 ) [ .051 ]
            10
            76 ( . 091 ) [ . 103]
            85
            ( .099 ) [ . 110 ]

            163
            n
            c
            = 3
            a
            nl=3
            n i =4
            nl=5
            .01
            —
            ( ) [ .016]
            16
            ( .008) [ .023]
            21
            ( .004 ) [ .012 ]
            .025
            12
            ( .016) [ . 047 ]
            15
            ( . 023 ) [ . 047]
            19
            C.023)[.047]
            .05
            11
            (.047 )[ .094 ]
            14
            ( .047 ) [ .094 ]
            18
            ( .047 ) [ .078 ]
            .10
            10
            (.094)[.188]
            13
            (. 094 ) [.156]
            17
            (. 078 ) [ . 121 ]
            =6
            ni=7
            n j =8
            .01
            26
            ( .006 ) [.011]
            32
            ( .006)[.012]
            38
            (.010) [ .015 ]
            .025
            24
            ( .023 ) [ .039 ]
            30
            ( .020) [ .030 ]
            36
            ( .023 ) [ . 032 ]
            .05
            23
            (.039)[.060]
            28
            (.046)[.065]
            34
            (.045)[.062]
            .10
            21
            (.092 ) [ . 129 ]
            26
            ( .090) [ . 120 ]
            32
            (.081)[ . 104 ]
            nl=9
            n x = 10
            n ^ = 11
            .01
            4 6
            ( .008) [.Oil]
            54
            ( .008) [.Oil]
            63
            ( .008)[ .010]
            .025
            43
            ( . 023 ) [ .031]
            51
            (.021 ) [ . 027 ]
            59
            (.023)[.029]
            .05
            41
            (.041)[.054]
            48
            ( .045 ) [ .056 ]
            56
            (.044 ) [ .053]
            . 10
            38
            (.086)[. 107 ]
            45
            (.084 )[.101 ]
            52
            (. 090)[. 106]
            nj-12
            n ^ = 1 3
            = 1 4
            .01
            72
            ( .009) [ .011]
            82
            ( .009 ) [ .011]
            93
            ( . 009 )[.011]
            .025
            68
            (.023 ) [ .028 ]
            78
            ( .021 ) [ .025 ]
            88
            ( .022 ) [ . 026 ]
            . 05
            64
            (.048)[.057]
            73
            (. 049) [ . 056 ]
            83
            ( .046) [ .053]
            . 10
            60
            (.090 )[.104 ]
            68
            ( .097 ) [ . 1 10 ]
            77
            ( .098)[.110]
            n i = 1 5
            .01 1 04( .009)[.011]
            .025 98 ( .025) [ .028]
            .05 93 (.048)[.055]
            . 10 87 (.095)[ .105]

            164
            n
            c
            = 4
            a
            ni=4
            nl=5
            n i = 6
            01
            20
            (.004 ) [ .012 ]
            24
            (.006 ) [ .012 ]
            29
            ( .006 ) [ .012]
            025
            18
            (.023)[.047]
            22
            ( . 023 ) [ .041 ]
            27
            ( .021 ) [. 033]
            05
            17
            (.047 )[ .082 ]
            21
            ( .041 ) [ .066 ]
            26
            ( .033 ) [ . 052 ]
            10
            16
            (.082)[.129]
            20
            (. 067 )[ . 101 ]
            24
            ( . 076 ) [ . 106]
            n 1 =7
            n ^ = 8
            nl = 9
            01
            35
            ( .006)[.010]
            41
            ( . 008) [ .013]
            48
            ( .010)[.014]
            025
            33
            (.017 ) [ .026 ]
            39
            ( .019) [ .028]
            46
            (.019)[.026]
            05
            31
            (.0397[.055]
            37
            ( . 038 ) [ .052]
            43
            ( .046 ) [ . 059]
            10
            29
            (. 075 ) [ . 101 ]
            34
            ( .089 ) [.112]
            40
            ( .092 ) [ . 1 13 ]
            n x = 10
            n : = 11
            n j = 1 2
            01
            56
            (.010) [ .013]
            65
            (.009 ) [ .012 ]
            74
            (.010)] .013]
            025
            53
            ( .023 ) [.030]
            61
            (.025)[.031]
            70
            ( .024 ) [ . 029]
            05
            50
            (.048 )[ .060]
            58
            (.046 )[ .056 J
            67
            ( .042 ) [ .050]
            10
            47
            (.088)[.105]
            54
            ( . 094 )[.109]
            62
            (.093)1.107]
            n j = 1 3
            = 1 4
            n i = 1 5
            01
            84
            (.010)1.012]
            95
            (. 009 ) [.Oil]
            106(.010)[.012]
            025
            80
            ( . 022 ) [ .026 ]
            90
            ( .023 ) [ .027 ]
            10 1 ( .022 ) [ . 025]
            05
            76
            ( .043 ) [.050]
            85
            ( . 048) [.055]
            95
            ( .049) [. 056]
            10
            70
            ( .099 ) [ .112]
            80
            ( .089 ) [ . 100]
            89
            ( .096 ) [ . 107]

            165
            n
            c
            = 5
            a
            nl=5
            n ^ =6
            nl = 7
            01
            28
            ( .006) [ .012 ]
            33
            ( .006) [ .010]
            38
            ( .009 )[.014]
            025
            26
            ( .021 ) [ .034]
            31
            (.017) [ . 027]
            36
            ( .021 ) [.030]
            05
            25
            ( .034 )[ .054]
            29
            ( .041 ) [ . 059 ]
            34
            ( .043 ) [ . 059 ]
            10
            23
            ( . 079 ) [ . 112]
            27
            (. 083 ) [ . 112]
            32
            (.079)[.103]
            n ^ =8
            nl=9
            rij = 1 0
            01
            45
            ( . 007 )[.010]
            52
            ( . 008 ) [.Oil]
            60
            (. 008) [.010]
            025
            42
            ( .022 ) [ .030]
            49
            ( .021 ) [ .028]
            56
            ( .024 ) [ .031 ]
            05
            40
            (.041)[.054]
            46
            (. 047 ) [ . 059]
            53
            (. 048) [. 059]
            10
            37
            (.089)[.111]
            43
            (.091)[.110]
            50
            ( .086) [ . 103]
            tij-11
            n ^ = 1 2
            n x = 1 3
            01
            68
            ( .009 ) [ .012]
            78
            ( .008)[ .010 ]
            87
            ( .010) [ .012]
            025
            64
            ( . 025 ) [ .031]
            73
            ( .024 ) [ . 029 ]
            83
            ( . 022 ) [. 026]
            05
            61
            (.046 )[ .055 ]
            69
            ( .049 ) [ .058 ]
            78
            ( .049 ) [ . 057 ]
            10
            57
            (.091)[.106]
            65
            (. 090)[ . 103]
            73
            ( . 097 ) [ . 109]
            n y = 1 4
            n ^ = 1 5
            01
            93
            ( . 009) [.Oil]
            1 09(.010)[.012]
            025
            93
            (.022 )[ . 026 ]
            103(.025)[ . 029 ]
            05
            88
            (.047)[.053]
            98
            ( . 048) [.054]
            10
            82
            (.097 )[ . 108 ]
            92
            ( .094 ) [ . 104]

            a
            01
            025
            05
            10
            01
            025
            05
            10
            01
            025
            05
            10
            01
            025
            05
            10
            166
            n
            c
            = 6
            n j =6
            nl=7
            n j =8
            37
            ( .009)[ .014]
            43
            ( .007 ) [ .01 1 ]
            49
            ( .008)[ .012 ]
            35
            (.021)[.031]
            40
            (.023)[.032]
            46
            (.023)[.030]
            33
            ( .044 ) [ .061 ]
            38
            ( .044) [ . 058 ]
            44
            ( .040 ) [ .052 ]
            31
            (.082)[.108]
            35
            (. 097 ) [ . 122]
            41
            ( . 084) [ . 104]
            nl=9
            n ^ = 1 0
            n x = 11
            56
            ( .008) [.011]
            64
            (. 008 ) [ .011]
            72
            (.009) [ .011 ]
            53
            (.021)[.027 J
            60
            ( .023) [ .029]
            68
            ( .024 ) [ .029 ]
            50
            (.045)[.056]
            57
            ( . 045 ) [ . 055]
            65
            (.043)[.051]
            47
            (.084 ) [ . 102 ]
            53
            (.095 )[ .111]
            60
            ( .098) [ .113]
            n : = 1 2
            n ^ = 1 3
            n ^ = 1 4
            81
            (.010)[.012]
            91
            ( .009 ) [ .012]
            102( .009) [.Oil]
            77
            ( . 022 ) [. 027]
            86
            ( .024)[ .029]
            96
            ( . 024 ) [ .028]
            73
            ( .046 )[ .054 ]
            82
            ( .046 ) [ .053 ]
            91
            ( .050) [ . 056 ]
            68
            ( . 096 ) [.109]
            77
            (.090)[.101]
            86
            ( . 090) [ . 101 ]
            = 1 5
            11 3( .009)[ .011]
            107 ( .023 ) [ .027 ]
            102( .045 )[ .050]
            95 (.097)1.107]

            167
            n
            c
            = 7
            a
            n 1 =7
            n j =8
            nl=9
            .01
            48
            ( .008) [ .012]
            54
            ( .009 ) [ .012 ]
            61
            ( .008) [ .011]
            .025
            45
            ( . 023 ) [ .031 ]
            51
            ( . 022 ) [ .029]
            58
            ( . 020 ) [. 025 ]
            .05
            43
            ( .042 ) [ .054 ]
            48
            ( .048 ) [ .060]
            54
            ( .050) [ .061]
            . 10
            40
            (.087)[.108]
            45
            ( .091)[. 1 11 ]
            51
            ( . 090) [ .107]
            n ^ = 1 0
            n1 = 1 1
            n L = 1 2
            .01
            69
            ( . 008) [.010]
            77
            ( . 009 ) [.Oil]
            86
            ( . 009)[.011]
            .025
            65
            ( .021 ) [ .026]
            73
            ( .021 ) [ .026 ]
            81
            ( .024) [ .029 ]
            .05
            61
            ( . 049) [. 059]
            69
            ( . 046 ) [ . 054]
            77
            (.048)[.055]
            . 10
            57
            ( .098)[.114]
            64
            (.100 ) [ . 1 15 ]
            72
            ( .09 7 ) [ . 1 10 ]
            n x = 13
            n ^ = 1 4
            n j = 1 5
            .01
            96
            (.009 )[.010 ]
            106 ( .009 )[.Oil]
            117( . 0 1 0 ) [ .011 ]
            .025
            91
            (.022)[.025]
            101 (. 022 ) [ .025]
            1 1 1 (. 024 )[. 027 ]
            .05
            86
            (. 047 ) [ . 053 ]
            96
            ( .044 ) [ .050]
            106( .045 ) [ .051 ]
            . 10
            81
            (. 090)[.101 ]
            90
            (. 090)[ . 100]
            99
            (.096)[.106]

            168
            n
            c
            = 8
            a
            n ^ =8
            n 1= 9
            n j = 1 0
            .01
            60
            (.008)[.Oil]
            67
            ( .008 ) [ .010]
            74
            ( .009) [ .012]
            .025
            57
            ( .020) [ . 026]
            63
            ( . 022 ) [ .028]
            70
            (.023)[.028]
            .05
            54
            ( .042 ) [ .052]
            60
            ( .043 ) [ .052 ]
            67
            ( .042 ) [ .050]
            . 10
            50
            ( .093) [ .111]
            56
            ( . 090) [ . 106]
            62
            (. 096)[. Ill]
            n x = 1 1
            = 1 2
            n i = 1 3
            .01
            82
            ( .010) [ .012]
            91
            ( .010)[.012]
            101 (. 009 ) [ .011 ]
            .025
            78
            ( . 022 ) [ .027 ]
            86
            ( .025 ) [ . 029 ]
            96
            ( .022) [ . 025 ]
            . 05
            74
            ( .046 ) [ .054]
            82
            (. 047 ) [ .054]
            91
            ( . 046 ) [ . 052 ]
            .10
            69
            (.097 ) [ . 11 1 ]
            77
            ( .093 ) [ . 105]
            85
            ( . 09 7 ) [ . 108 ]
            n j = 1 4
            nx = 1 5
            01
            *—(
            o
            •
            o
            o
            •
            V-/
            f ^
            » <
            r—i
            122( .010) [ .011 ]
            025
            105 ( .025 ) [ .029 ]
            1 16 ( .023 ) [ .027 ]
            05
            1 00(.049) [ . 055]
            110 C. 049)[.055]
            10
            94 ( . 096 ) [ . 106 ]
            104( .092 ) [ . 101 ]

            169
            n
            c
            = 9
            a
            nl=9
            o
            II
            c
            n i = 11
            .01
            73
            ( .009)[ .012]
            81
            ( .008)[ .010]
            89
            ( .008)[ .010]
            .025
            69
            (.024)[.029]
            76
            ( .024 ) [ .029]
            84
            ( . 022 ) [ . 027]
            .05
            66
            ( .043 ) [ .052]
            72
            ( .049 )[ .057 ]
            80
            ( .044) [ .051]
            . 10
            61
            ( .099) [.115]
            68
            ( . 090 ) [ . 104]
            75
            (.090)[.103]
            n i - 1 2
            n ^ = 1 3
            n i = 1 4
            .01
            97
            ( .010) [ .012]
            1 07(. 009 )[ .011]
            1 17 ( . 009 ) [.Oil]
            .025
            92
            ( .024 ) [ .028 ]
            10 1 ( .024 ) [ .028]
            1 1 1 ( .024 ) [ .027 ]
            .05
            88
            ( .045) [ .051 ]
            96
            ( . 049 )[ .056]
            106( . 045) [ .051 ]
            .10
            82
            (.098)[.110]
            91
            (.090 )[ . 100]
            99
            ( .098 )[ . 108]
            n x = 15
            .01 1 28( . 009 ) [ .011 ]
            .025 1 22( .022 ) [ .025 ]
            .05 11 6( .046 ) [ .051 ]
            . 10 109( .093 ) [ . 102 ]

            170
            o
            t—H
            II
            CJ
            e
            a
            n x = 10
            nj-ll
            n x = 1 2
            01
            88
            (.008)[ .010 ]
            96
            ( .008)[ .010]
            104( .010)[ .012]
            025
            83
            (.023)[.027]
            91
            ( .021 ) [ . 025]
            99 ( .023) [.026]
            05
            79
            (.045)[.053]
            86
            ( .047 ) [ .054 ]
            94 (.047)1.053]
            10
            74
            (.093)[.106]
            81
            (.092)[.104]
            88 (.098) [ .109]
            n j = 1 3
            n i = 1 4
            n ^ = 1 5
            01
            1 1 4 ( . 009)[ .010]
            1 24( .009 )[.010]
            1 35(.009 )[ .010]
            025
            108( .023 )[ .026]
            117 ( .025 )[ .028]
            1 28( .023 )[ .026]
            05
            103( .045 ) [ .051]
            1 12( . 046 ) [. 052 ]
            1 22 ( . 046 )[.051]
            10
            96
            ( . 0 9 9 ) [ .110]
            105 ( .097 ) [ . 107 ]
            1 1 5 ( .092 ) [ .101]

            APPENDIX 2
            THE MONTE CARLO PROGRAM
            The Monte Carlo program listed in this appendix, was written
            for this research using fortran (FORTXCG, i.e., SYSTEM/370
            fortran H extended (enhanced)). Computing was done utilizing the
            facilities of the Northeast Regional Data Center of the State
            University System of Florida, located on the campus of the
            University of Florida in Gainesville. It used available IMSL
            subroutines (e.g., GGUBS, GGNSM, RANK, ..., etc.) whenever
            possible. The single precision version of this library was used.
            171

            nnn o non non onnn non
            172
            DCOBL£ PRECISION DSEED,DSEED2
            DIMENSION POWERS (45) ,X1(4C) ,X2(40) , SIGMA (3) ,X(4Q,2) ,
            IIRWVEC ( 40) , RW V EC (4 0) ,WKVE C (2) f DFTST (3) , D F M ST (3) ,C(4Q)
            REAL LI (3) , L2 (3)
            INTEGER REPS,N,NS,NNS,NPROb, NCD,NOCENS (41,41) ,D(40)
            C
            C THIS PROGRAM RUNS A MONTE CARLO FOR A SAMPLE SIZE UP TO 40
            C
            C OBTAIN PARAMETERS JOE THIS RUN Or THE MONTE CARLO
            C
            CALL I NIT (REPS,N,NS,NNS,XMU,L1,L2,SIGMA,DSEED,DSEFD2)
            DC 5 1=1,NNS
            5 POWERS (I) =0,0
            NFEOB= 0
            NCD=0
            C NPS03 IS THE * OF SAMPLES WITH NO UNCENSORED PAIRS, WHILE
            C NCD IS THE # OF TIMES CDTST HAS A N5G. VARIANCE ESTIMATE
            C
            DC 10 1=1,41
            DC 10 J=1,41
            10 NOCENS(I,J)=0
            START THE REPLICATIONS
            CO 100 IREPS=1,REPS
            GENERATE AN N RANDOM BIVARIATE NORMALS
            WITH COVARIANCE MATSIXSIGMA
            CALL SAMPLE(DSEED,DSEED2,N,SIGMA,W KVEC,I REPS,X,X1,
            IX 2 , C)
            NC» TO PREFORM THE CENSORING ON THE RANDOM VARIABLES
            CALL CENSOR(X1,X2,C,N,D)
            CALCULATE THE TEST STATISTICS
            CALL DPST AT(X1,X2,D,N,XMU,L1,L2,NPR03,NCCENS,DFTST,
            IDFMST)
            IF (DFTST (1) .EQ. 999.9) GO TO 100
            CALL CDS TAT (X1,X2,D, N , CDTS T, CDT ST2 , CDTST 3, NCD)
            COLLECT SUMMARY AND POWER STATISTICS
            CALL POWER(NS,NNS,DFTST,DFMST,CDTST,CDT3T2,CDTST3,
            IPOWERS)
            100 CONTINUE
            C

            17 3
            C PEINT GDI THE RESULTS
            C
            WHITE(6,500) NPROB
            500 FORMAT(•-','THE NUMBER OF SAMPLES DISCARDED, DUS
            ¿'TO N1 = 0 WAS', 1 X, 13)
            WRITE (o,502) NCD
            502 FORMAT (
            bO 1
            600
            o 1 0
            d50
            i _ i
            •THE NUMBER OF SAMPLES WHICH HAD A '
            505
            510
            515
            520
            525
            530
            ai' NEGATIVE VARIANCE ESTIMATE FOR CD WAS • , 1 X, 18)
            WRITE(6,505)
            FORMAT 'TtlE DISTRIBUTION OF CENSORING: THE ROWS',
            ¿' ARE FOR TYPE 2 OR 3 AND THE COLUMNS FOR TYPE 6' //)
            DO 510 1=1,41
            WRITE (6,515) I, (NOCENS (I, J) , J=1,26)
            FORMAT(' ' ,12, •:•,4X,26I4)
            IF (N .LT. 26) GO TO 530
            DO 520 1=1,41
            WHITE (6,525) I, (NOCENS (I, J) , J = 27,41)
            FORMAT(' ',12,':',4X,25I4)
            CONTINUE
            NPRCP=0
            NX=N+1
            DO 600 1= 1 , NX
            DC 601 J = 1,NX
            NPROP=NPROP+ ( ( (I- 1) + {J- 1) ) * NOCENS (I, J) )
            CONTINUE
            AVG= (FLOAT (NPROP) ) / (FLOAT (REPS) )
            WRITE(6,610) AVG
            FORMAT('-','IHE AVERAGE NUMBER OF OBSERVATIONS',
            a)' CENSORED IS: ',F10.5)
            WRITE(6,550)
            FORMAT ('-', 15X, 'FINAL RESULTS', /, 20X,
            'STAT 4 1: DP 1ST 1 ' , / 20 X, 'STAT *2: DFTST2 ' ,
            / 20X, ' STAT #3: DFTST3 • , / 20X,'STAI #4: DF LIST 1 ' ,
            / 20X,'STAT #5: DFMST2', / 20X,'STAT ¿ó: DFMST3',
            / 20 X, 'STAT 47: CDTST (WITHOUT MEAN)',
            / 20 X, ' STAT 43: CD TST2 ('WITH MEAN)',
            / 20X,'STAT ¿9: CDTST3 (ASYMP WITH MEAN)')
            CALL ÃœSWSMC POWER MAT RIX/iiEJSCTS * ,20 , POW ERS , NS , 1)
            STOP
            END
            SUBRCUTINE IN IT (REPS,S,NS,NNS,XMU,L1,L2,SIGMA,DSEED,
            ¿D5EED2)
            C
            C THIS SUBROUTINE READS THE NUMBER OF REPLICATIONS (REPS),
            C THE SAMPLE SIZE PER RUN (N) , THE POPULATION COVARIANCE
            C MATRIX (SIGMA) , AND THE VECTORS LI AND 12 FOE THE STATISTIC
            C DFT ST. IT STORES SIGMA IN SYMMETRIC STORAGE MODE (IKSL) -
            C
            DOUBLE PRECISION DSE ED, DSEED2
            SEAL L1 (3) ,L2 (3)
            DIMENSION SIGMA (3)
            INTEGER REPS

            r¡ n n n n n n n n n o n n non
            174
            Ii£PS= 1 000
            N=25
            NS= 9
            SHS=(NS*(N 3 + 1))/2
            L 1 (1)=1.0
            L2 (1)= 1.0
            LI (2) =2.0
            L2 (2) = 1. 0
            L 1 ( 3) = 1.0
            L2(3) =2. 0
            DSEED=335768.DO
            DSE ED2= 672344.DO
            WRITE (6,100) BSPS, N
            100 FORMAT ,15X,'# HEPS =' ,14, 10X , 'SAMPLE SIZE (N) =',
            d)I2)
            new TO LEAD IN THE COMMON LOCATION PARAMETER (XNU)
            xau=o.o
            NOW TO ENTER THE CGVARI ARC 2 MATRIX (SIGMA)
            R HC=.2
            V ABX1 = 1.0
            V A 2X2=3. 0
            SIGMA(1)=VAN X1
            SIGMA(2)= RHÜ +(SQRT(VARX1)) *(SQRT (V A R X 2))
            SIGMA(3)=VARX2
            ECHO CHECK
            WRITE (6, 105)
            105 FORMAT('0'// 10X, • BIVARIATE NORMAL DISTRIBUTION
            ¿•GENERATED')
            CALL USWSM ('COV. MATRIX SIGMA17,SIGMA,2,2)
            RETURN
            END
            SUBROUTINE SAMPLE(D3EED,D3E3D2,N,SIGMA,WKVEC,IREPS,
            a)X,X1,X2,C)
            THIS SUBROUTINE GENERATES THE NX2 RANDOM VECTOR OF
            OBSERVATIONS
            DOUBLE PRECISION DSEED,QSEED2
            DIMENSION X(N,2) rWKVEC{2) , SIGMA (3) ,X1 (N) ,X2 ( N) ,
            «U (40) ,C (N)
            INTEGER N,IER
            CALL THE IMSL NORMAL RANDOM VECTOR SUBROUTINE
            WKVEC (1) = 1. 0
            IF (IREPS .EQ. 1) WKVEC (1) =0.0

            n n n n n n non
            17 5
            CALL GGNSM(DSEED,N,2,SIGMA,N,X,WKVEC,I EE)
            If (IEE . N E. 0) aaiTE(6,500) IREPS,I2D
            500 FORMAT ( ' , 1 OX, 'GGNSM ERROR, REPLIC AT ICN= * , 14 ,
            <2'IER=',I4)
            DO 25 1=1 ,N
            X1 (I)=X(I, 1)
            X2(I)=X(I,2)
            25 CONTINUE
            C
            C LOW TO GENERATE THE CENSORING DISTRIBUTION
            C
            C FIRST GENERATE A SAMPLE 0? N iJ ill FORM (0,1) h V' S
            C
            CALL GGUBS ( DSEED2, N, U)
            NCS TC GENERATE THE CENSORING RANDOM VARIABLES
            DO 28 1=1,N
            ¿8 C (I) =ALCG (o, S975*D (I) )
            28 C (I) = ALGG (b. 8371*U (I) )
            28 C (I) =ALOG (b, 3389*0 (I) )
            RETURN
            END
            SUBROUTINE BAN K (NU,Z,SZ,IR#VEC,RKV EC)
            THIS SUBROUTINE CALCULATES THE VECTOR RANKS
            DIMENSION Z (NU) ,EZ (NU) ,ISWVEC (NO) ,RWVEC ( NU)
            EFS=0.00000001
            CALL NHRANK (Z, NO , E PS ,1RH VEC , P.« VEC , KZ , S 2, S3)
            RETURN
            END
            SUBROUTINE CENSOR(XI,X2,C,N,D)
            C
            C THIS SUBROUTINE CENSORS THE DATA AND CREATES A VECTOR D OF
            C THE TYPE OF CENSORED PAIR, A PARTICULAR PAIS IS,
            n
            v*
            DIMENSION XI (N) ,X2 (N) ,C (N)
            INTEGER D(N)
            C
            DO 6 1=1,N
            IF (XI (I) , NE. X2 (I)) GO TC 100
            IF (X2 (I) .EE. C (I) ) D (I) =1
            IF (X2 (I) , GT, C (I) ) GO TO 102
            100
            GO TO 6
            IF (X2(I) , LE,
            XI (I) )
            GO TO
            4
            IF (X 2 (I) .GT.
            c (I) )
            GO TO
            105
            10 5
            D (I) = 1
            GO TO 6
            IF (XI (I) ,LE,
            C (I))
            GO TO
            1 10
            10 2
            D(I)=4

            176
            X1 (I)=C (I)
            X2(I)=C(I)
            GO TO 6
            lie D (I) =2
            X2 (I)=C(I)
            GO TO 6
            4 If (X 2 (I) .LE. C (I) ) GO TO 115
            D(I)=4
            X1 (I)=C (I)
            X2 (I) =C (I)
            GO TO 6
            115 IF (XI (I) ,L£. C(I)} GO TO 120
            D (I) =3
            X 1 (I) =C (I)
            GO TO 6
            120 D (I) = 1
            6 CONTINUE
            C
            2ETURN
            END
            SUBROUTINE ESTMU (X 1 , X 2, D , X , E MU)
            C
            C THIS SUBROUTINE CALCULATES THE COMBINED SAMPLE HEDIAN
            C USING THE KAPLAN MEIER ESTIMATOR WITH SMOOTHING (EMU)
            C AND WITHOUT SMOOTHING (SMD).
            C
            DIMENSION X1 (N) ,X2 (N) ,RVEC (8 0) ,RY(8Ü) , Y (80) , YY (80) ,
            áliVEC (30) ,S (80)
            INTEGER D (N) , DD (30) , DYY (80)
            K= 1
            DO 10 1=1, N
            IF (D (I)-3) 20,20, 1 1
            20 J = 2*K
            JJ=J-1
            Y (J ) = X2 (I)
            Y (JJ) =X 1 (I)
            IF (D (I)-2) 22,24, 26
            22 DD (J) = 1
            DD(J-1) =1
            GO TO 28
            24 DD (J) =0
            DD(J-1)=1
            GO TO 28
            26 DD(J) = 1
            DD (J-1) =0
            GO TO 28
            11 J=2*X
            JJ=J-1
            Y (J) =X2 (I) +0. 00000 01
            Y (J J) =X 1 (I)
            DD (J) =0
            DD (JJ) =0
            28 K=K +1

            n n n
            177
            10 CONTINUE
            C
            CALL HANK (Jf Y ,£Y,IWVSC,RVSC)
            DC 780 1=1,J
            L=IFIX (RY (I) )
            YY(L)=Y (I)
            780 E Y Y {L) =DD (1)
            C
            C COMPUTING THE KAPLAN MEIER ESTIMATE CF THE SURVIVAL
            C FUNCTION
            C
            XJ=FLCAT (J)
            S (1) = ( (XJ-1. 0) /XJ) ♦♦DYY { 1)
            DC 500 1=2,JJ
            XI=FLCAT(I)
            500 S(I)=S (1-1) *(( (XJ-XI) / (XJ-XI+ 1,0) ) **EYY(I) )
            S (J) =0. 0
            c
            C NOW' TO CALCULATE THE MEDIAN ESTIMATES
            C
            DO 550 1=1,J
            IF (S (I)500000) 530, 540, 550
            540 2KU = YY (I)
            GO TO 560
            830 E1 = Y Y (I)
            E2=YY(1-1)
            EMU=E1-((E1-E2)*{> 500000-S (I))/(S (I-1)-S (I)))
            SMU= (E1+E2)/2. 0
            30 TO 560
            550 CONTINUE
            560 CONTINUE
            C
            RETUEN
            END
            S ÃœBEOUTINE DFSIAT(XI,X2,D,N,XMU,L1,L2,NPPC3,NOCENS,
            áDFTST,DFMST)
            C
            C THIS SUBSQUTIN3 COMPUTES THE TEST STATISTIC CALLED DFTST
            C (DFMST), WHICH IS THE COMBINATION OF TWO SIGNED SANK
            C STATISTICS DFTST(I) = L1(I)*WILCOXON + L2 (I)*WILCGXCN
            C
            SEAL L 1 (3) ,L2 (3)
            DIMENSION XI (N) , X2 (N) ,2 (40) , T2 3 (40) , PHI (40) ,RMVEC(40) ,
            iáDFMST (3) , DFTST (3) ,RZ (40) ,T (3) , VAF.T (3) ,SDT (3) ,RT23 (40) ,
            d)K ZZ (40) , IH W V E C (40) ,3AM (40) ,ZZ{40) ,2PHI(40) , IT (3) ,21(3)
            INTEGER D(N),NOCENS(41,41)
            N 1 = 0
            NC=0
            C
            NOW TO ESTIMATE THE VALUE OF MU (EMU) USING THE PRODUCT-
            LIMIT ESTIMATOR BASED ON THE ENTIRE SAMPLE
            CALL ESTKU(XI,X2,D,N,EMU)

            n n n non onnn nnn non nnn
            178
            C
            C THE FIESI PAST OF THE SUBROUTINE, UTILIZES WHAT TYPE OF
            C PAIS (X1,X2) IS: TYPE UNCENSG2ED,TYPE 2= X2 CENSORED,
            C TYPE 3= X1 CENSO RED,TYPE 4=CCTH CENSORED
            C AND PLACES T HE UNCENSORED CALCULATIONS IN T1 , WHILE THE
            C TYPE 2 OS 3 CENSORED C VALUES GO INTO THE VE'CTCH T23
            C
            DO 6 1=1,N
            IF (D (I) . EQ. 4) GO TO 6
            IF (D (I)-2) 5,3,4
            A PAIS WILL GO TO 3, I? IT IS A TYPE 2 PAIR
            J NC= NC +1
            GAM (NC) =1.0
            T23(NC)=X2 (I)
            GO TO &
            A PAIR WILL GO TO 4, I? IT IS A TYPE 3 PAIR
            4 NC=NC+1
            GAM(NC)=0.0
            T23 (NC) =X 1 (I)
            GO TO 6
            A PAIS WILL GO TO 5, I? IT IS A TYPE 1 PAIR
            d N1= N1+ 1
            Z (N1) = (AI3S ( (X2 (I) ) -XMU) ) - (ADS ( (X 1 (I) ) -XMU) )
            ZZ (N 1) = (AES ( (X2 (I) ) -EMU) ) - (ABS ( (X 1 (I) ) -2HU ) )
            PHI (N1) = (SIGN ( 1. 0,Z (N 1) ) /2.0) +0. 5
            ZPHI (N1) = (SIGN (1 .0,ZZ (N 1) )/2.0) + 0.5
            Z (N 1) = ABS IZ (N 1) )
            ZZ(N1)=AES(ZZ(N1))
            CONTINUE
            IF (N1 . EQ. Ü) GO TO 100
            TO INSERT WHAT TYPE OF CENSORING OCCURRED INTO THE MATRIX
            NOCENS (NN4=#TYP3 4 PAIRS + 1, NNC=*TYFE 2 OR 3 RAIDS + 1)
            NN 4 = 1 + N-N1-NC
            NNC=NC+1
            NOCENS(NNC,NN4)=NOCENS(N NC,N N 4)+1
            CALCULATING THE ABSOLUTE RANKS FOR THE N1 UNCENSCRED OBS.
            CALL RANK(N1,Z,SS,IRWVEC,EWVEC)
            CALL RANK (N 1, ZZ , 3ZZ, IB'rf VEC , RWVEC)
            NOW TO CALCULATE THE RANKS OF THE CS FOR THE TYPE 2 AND 3
            IF (NC , NE, 0) GO TO 2 5
            WRITE (6,23) NC
            23 FORMAT(•-«,'THERE ASE NO TYPE 2 OR 3 CENSORED ',

            n n
            179
            it 'OBSERVATIONS, NC = ',13)
            GC TO 28
            2b CONTINUE
            CALL RANK (MC#T23,ST 23, Ia*VEC,3RVEC)
            C
            28 CONTINUE
            C
            C NOW TO CALCULATE THE WILCOXON TlEE STATISTICS, SUN 1 AND
            C SUHC, AND THE CORRESPONDING EXPECTED VALUES AND VARIANCES
            C
            suai=o,o
            s u a 11=0. o
            DO 30 1=1, N1
            SUE1=SUM 1+(PHI (I)*HZ (I))
            SUH11=SUM11+ (ZPÍ1I (I) *RZZ (I) )
            30 CONTINUE
            VAR 1 = (FLOAT(N1*(N1 + 1)*((2*N1) + 1)))/24.C
            E1 = (FLOAT (N1*(H1 + 1)) )/4.0
            C
            SUMC=Q.0
            IF (NC . NE , 0) GC TO 3 3
            V AHC=0.0
            EC=0.0
            . GC TO 35
            33 CONTINUE
            DO 34 1=1, SC
            SUMC=SUMC+ (GAM (I) *RT23 (I) )
            34 CONTINUE
            VARC= (FLOAT(NC*(NC + 1)*((2*NC) + 1) ))/24.0
            EC= (FLOAT (NC* (NC+1)} )/4.0
            C
            35 CONTINUE
            C
            NOW TO CALCULATE THE DFTST
            DO 38 1=1,3
            T (I) = (LI (I) *SUM1) + (L2 (I) *SUKC)
            TT (I) = (L 1 (I) *SUM11) + (L2(I) *SUMC)
            El (I) = (LI (I) *E 1) + (L2 (I) *EC)
            VART (I) = ( (L1 (IJ **2) *VAR1) + ( (L2 (I) **2) +VAHC)
            SDT (I) =SQBT (VAST (I) )
            EFTST (I) = (T (I) -ET (I) ) /SDT (I)
            DFHST (I) = (TT (I) -ET (I) ) /SDT (I)
            3e CONTINUE
            C
            GO TO 47
            100 CONTINUE
            C
            C THERE IS A PROBLEM, N1=0, THUS THE SAMPLE IS NOT GOING TO
            C BE USED IN THE POWER STUDY. THE TEST STATISTIC WILL 5L
            C SET TO 999.9 WHICH WILL BE USED AS AN INDICATOR.
            C
            DO 102 1=1,3
            DFMST(I)=999.9

            U U U U {J CJ
            180
            102 DFTST(I)=999.9
            WHITE(b,104)
            104 FORMAT (, '*** PROBLEM, N1=0, THE SAMPLE HILL HOT
            NPfiuE=NPROB+1
            C
            47 RETURN
            END
            SUB SOUTINE CDSTAT (X1,X2,D, N, CDTST, CDTST2 , CDTST3 , NC D)
            C
            c
            C THIS SUBROUTINE CALCULATES THE CONCORDANT - DISCORDANT
            C TYPE STATISTIC (CDTST).
            C
            DIMENSION X1 (N) ,X2(N) , XI (40) ,Y2(40)
            INTEGER D(N) ,A (40,40),5 (40,4 0) ,SUKCD,SUMA,SUHS,SUM1,
            üSUE2,SUa3fSOMGG
            DO 1 1=1,N
            DC 2 J=1,N
            A (I, J) =0
            2 E (I, J) =0
            1 CONTINUE
            DO 5 1=1, N
            Y 1 (I) = X1 (I) + X2 (I)
            5 Y2 (I) =X 1 (I)-X2 (I)
            NOW TO CALCULATE THE A(I,J) AND B(
            J) MATRICES
            200
            ¿20
            230
            20
            10
            NN=N-1
            DO 10 1=1,NN
            11=1+1
            DO 20 J=II,N
            IF (Y 1 (I) - Y 1 (J) ) 200,230,220
            IF (D (I) ,EQ, 1) A (I, J) = 1
            GO TO 230
            IF (D (J) .EQ. 1) A (I, J) = -1
            IF (D (I) . EQ. 4) GO TO 10
            IF (D (J) .EQ. 4) GO TO 20
            IF (Y2 (I) -Y2 (J) ) 240,20,260
            IF (D(I)-3) 242,20,20
            IF (D (J) . EQ. 1) B (I, J) = 1
            IF (D (J) .EQ. 3) 3 (I, J) = 1
            GO TO 20
            IF (D (J) - 3) 262,20,20
            IF (D (I) .EQ. 1) B (I , J) =— 1
            IF (D (I) .EQ, 3) 3(I,J)=-1
            CONTINUE
            CONTINUE
            CALCULATING THE CONCORDANT-DISCORDANT STATISTIC (CDTST)
            SUMCD=0
            S U M A = 0

            181
            SU MG=0
            SUM1=0
            SUM2-0
            SUM3 = O
            SU MGG = 0
            lil = 0
            NN=N— 1
            DO 40 1=1,UN
            11=1+1
            DO 42 J=II,N
            SUKCD=SUMCD+(A(I,J)+B(I,J))
            5UMA=SUMA+IABS (A (I , J) * E (I, J) )
            IF (J . Efi. li) GO TO 45
            JJ=J+ 1
            DO 45 K=JJ,N
            SUM 1 = SUM 1+ (A (I, J) *A (I, K) *B (I, J)*B (I, K) )
            SUM2=SUM2+ (A (I, K)*A (J, K) *3 {I , K) * 8 (J , K) )
            SÃœM3=SUM3+ (A (I, J) *k (J,K) *E (I, J) *3 (J, K) )
            IN=IN+3
            45 CONTINUE
            42 CONTINUE
            40 CONTINUE
            VN = PLOAT (N)
            VNN=VN* (VN- 1,000)
            CD = 2. 000* (FLOAT (SUSCD) )/VNN
            S UMG= SUM1 + SUM2 + SUK3
            SUMGG=SUM1+SUM2+SUH3+SUMA
            AAA= (2,000* (FLOAT (SUM A) )/VNN) - (CD*CD)
            VIN=FLGAT (IN)
            VNNN = FLOAT (N* ( (N-1)**2))
            G= (FLOAT (SUKG) ) /VIN
            GGG= ( (2, 000* (FLOAT (SUMGG) ) )/VNNN) - (CD*CD)
            V A5CD=(4. 00 0*G)/VN
            VAE CD2=( (2, 0 Ou*AAA) + (4.00 0 * ( VN-2. 0 0 0) *GGG) )/VNN
            VASCD3=(4.000+GGG)/VN
            IF (VA3CD2 . Gl, 0,0) GO TO 55
            CD1ST=0.00
            CDTST 2=0.00
            CDT ST3=0,00
            NCD=NCD+1
            GO TO 65
            55 SDCD=SQET(VAECD)
            SDCD2=SQHT(VARCD2)
            SDCD3=SQRT (VARCD3)
            CDTST= (CD)/ (5DCD)
            CDTST2= (CD) / (SDCD2)
            CDTST3= (CD) / (SDCD3)
            65 CONTINUE
            EE1UKN
            END
            SUBROUTINE PO * EB (NS,MNS,DFTST,DFMST,CDTST,CDTST2,
            CÃœCDTST3, POKERS)
            C

            n n n
            182
            DIMENSION POWERS (NNS) ,REJECT (9) ,DFTST (3) ,DFM5T(3)
            C
            C NOTE: NS=NUMBEB OF STATISTICS CALCULATED
            C NNS= NS(NS+l)/2
            C
            C GIVE THE CRITICAL VALUES FOR THE TEST STATISTICS
            C
            ZCBIT=1. 645
            BZCRIT=-1.645
            CALCULATE THE POWERS
            10
            20
            C
            DC 10 1=1, NS
            REJ ECT (I) =
            IF (DFTST (1)
            IF (DFTST (2)
            IF (DFTST (3)
            if (df:ist(1)
            IF (DFMST (2)
            IF (DFMST (3)
            IF (CDTST , L
            IF (CDTST2 .
            IF (CDTST3 .
            DC 20 J = 1 ,NS
            JJ=J + 1
            = 0,
            0
            GE,
            GE.
            GE,
            GE .
            ZC ft IT)
            ZC R IT)
            ZCR IT)
            ZCSIT)
            ZCRIT)
            ZCRIT)
            REJECT ( 1)
            REJECT (2)
            REJECT (3)
            REJECT (4)
            REJECT (5)
            REJ
            , G E. ZCRIT) REJECT (6)
            D ZC SI T) RE J ECT (7) = 1
            3ZCRIT) REJECT (8) =
            REJECT (9) =
            LE
            LE, BZC3IT)
            = 1,0
            = 1.0
            = 1.0
            = 1.3
            = 1.0
            = 1.0
            , 0
            1.0
            1.0
            K= (J* (J- 1)/2+J)
            POWERS (K) =POWERS (K) + REJECT (J)
            IF (JJ .GT. NS) GO TO 20
            DO 20 I=JJ,N5
            K= (I* (I- 1) /2+J)
            POWERS (K) =POWSRS (K) * REJECT (I) ♦'REJECT (J)
            CONTINUE
            RETURN
            END

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            BIOGRAPHICAL SKETCH
            Laura Lynn Perkins was born in Harvey, Illinois, on August
            1, 1957. She moved to Lawndale, California, in 1961 and remained
            there until she moved to Titusville, Florida, in 1964. After
            graduating from Titusville High School in 1975, she enrolled at
            the University of Florida. Upon receiving her Bachelor of
            Science degree in mathematics in 1978, she entered the Graduate
            School and received her Master of Statistics degree in 1980. She
            expects to receive the degree of Doctor of Philosophy in-August,
            1984. She is a member of the American Statistical Association
            and the Biometric Society.
            Her professional career has included teaching various
            courses in the Statistics Department and consulting in the
            Biostatistics Unit of the J. Hillis Miller Health Center at the
            University of Florida. She has been the recipient of Graduate
            School fellowships, graduate assistantships, the Statistics
            Faculty Award and nominated for a Graduate Student Teaching Award
            during her academic career at the University of Florida.
            185

            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            Ronald H. Randles, Chairman
            Professor of Statistics
            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            Professor of Statistics
            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            John G. Saw
            Professor of Statistics
            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            I A
            Jonathan
            Professor
            J. S hu s ter
            of

            I certify that I have read this study and that in my
            opinion it conforms to acceptable standards of scholarly
            presentation and is fully adequate in scope and quality, as
            a dissertation for the degree of Doctor of Philosophy.
            Wayne [C. Huber
            Professor of Environmental
            Engineering
            This dissertation was submitted to the Graduate Faculty
            of the Department of Statistics in the College of Liberal
            Arts and Sciences and to the Graduate School, and was
            accepted for partial fulfillment of the requirements of the
            degree of Doctor of Philosophy.
            Augus t, 1984
            Dean for Graduate Studies
            and Research