A new nonparametric test for independence between two sets of variates

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A new nonparametric test for independence between two sets of variates
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xiii, 138 leaves : ill. ; 29 cm.
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English
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Gieser, Peter William, 1965-
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Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 134-137).
Statement of Responsibility:
by Peter William Gieser.
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Typescript.
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Vita.

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Full Text









A NEW NONPARAMETRIC TEST FOR INDEPENDENCE
BETWEEN TWO SETS OF VARIATES















By

PETER WILLIAM GIESER


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


1993





























) Copyright 1993

by

Peter William Gieser


































To my parents

and

the memory of William Trust












ACKNOWLEDGEMENTS


I would like to express my sincere gratitude to Dr. Ronald Randles, without whom

this work would never have been completed. His constant encouragement and opti-

mism were always sources of energy from which I could draw when things seemed

bleakest. I would also like to thank the members of my supervisory committee, as

well as all the faculty I have had a chance to get to know in the short time I have

been at the University of Florida. They will never realize the enormous impact that

their collective experience and knowledge has made on me. Special thanks go to Jane

Pendergast for going beyond the call of duty and being willing to help me in ways not

even related to statistics. To the many students whom I have met and become friends

with, I wish to acknowledge my pleasure in having had the privilege of knowing them.

I would especially like to thank Dan Bowling, who is probably one of the few people

who could have put up with me for so long. I consider him among the best friends I

have ever had. I am also indebted to Dr. James Kepner, who provided motivation via

his excitement about statistics and actually convinced me that I could get a Ph.D.

Finally, I would like to thank my family for their continual support and belief in my

ability to succeed.













TABLE OF CONTENTS




ACKNOWLEDGEMENTS ..................................................... iv

LIST OF TABLES ............................................. ...... ....... vii

LIST O F FIG U R ES ............................................................ viii

KEY TO SYMBOLS ........................................................ x

ABSTRACT ................................................................. xii

CHAPTERS

1 INTRODUCTION ................................................. 1

1.1 Bivariate Tests ............................................ 1
1.2 Multivariate Tests ........................................... .. 5

2 INTERDIRECTION QUADRANT STATISTIC ................... 9

2.1 Definition ................................................ 9
2.2 Null Distribution When (01,02) Is Known ................... 10
2.3 Null Distribution When (01,02) Is Unknown .................. 14

3 PITMAN ASYMPTOTIC RELATIVE EFFICIENCIES ............. 20

3.1 Introduction ............................................. 20
3.2 Model 1 ................................................. 21
3.3 Model 2 .................................................. 51

4 MONTE CARLO STUDY ........................................... 53

4.1 Methods ................................................. 53
4.2 Statistics Compared ......................................... 54
4.3 Results .................................................. 56







5 APPLICATIONS .............................................. ... 74

5.1 Analysis of Newborn Blood Gas Data ........................ 74
5.2 Analysis of Fitness Club Data ................................. 77
5.3 Analysis of Cotton Dust Data ................................. 78

6 CONCLUSION ........................................ ........ 86

6.1 Discussion .............................................. 86
6.2 Further Research ........................ ................. ... 87

APPENDICES

A CONVERGENCE RESULTS ........................................ 88

B CONTIGUITY ........................................................ 105

C SIMULATION STUDY ............................................. 108

REFERENCES ..............................................................134

BIOGRAPHICAL SKETCH .................................................138












LIST OF TABLES




Table Page

1.1 Pitman ARE's Reported by Farlie Under Bivariate Normality ....... 3

1.2 Pitman ARE's Computed by Konijn .................. ....... :3

4.1 Maximum Estimated Standard Errors for Empirical Power ......... 55

5.1 Newborn Blood Data ...................................... 76

5.2 Statistical Analysis of Newborn Blood Data .................... 78

5.3 Fitness Club Data ........................................ 79

5.4 Statistical Analysis of Fitness Club Data ....................... 79

5.5 Cotton Dust Data ......................................... 85

5.6 Statistical Analysis of Cotton Dust Data ....................... 85















LIST OF FIGURES


Figure

3.1 1/(1+ARE(-n logSJ, Q,))


3.2 1/(1+ARE(-nlog V, Q,; v = 0.1)) ...........................

3.3 1/(1+ARE(-nlog V, Q,; v = 0.5)) ...........................

3.4 1/(1+ARE(Q,, -nlogV; v = 1)) ............................

3.5 1/(1+ARE(Q,, -nlogV; v = 10)) ...........................

3.6 1/(1+ARE(-nlog V, Q,; df = 5)) ...........................


3.7 1/(1+ARE(Q,, -nlogV; df = 10)) .....

3.8 1/(1+ARE(Q,, -nlogV; df = 100)) ....

4.1 r = 1, n = 30, v = 0.1, reps = 2500 ......

4.2 r = 1, n = 30, v = 0.5, reps = 2500 ......

4.3 r = 1, n= 30, v= 1, reps = 2500 .......

4.4 r = 1, n = 30, v = 10, reps = 2500.......

4.5 r = 1, n = 30, df = 1, reps = 2500 .......

4.6 r = 1, n = 30, df = 5, reps = 2500 .......

4.7 r = 2, n = 30, v = 0.1, reps = 2500 .. ...

4.8 r = 2, n = 30, v = 0.5, reps = 2500 ......

4.9 r = 2, n = 30, v = 1, reps = 2500 .......

4.10 r = 2, n = 30, v = 10, reps = 2500.......

4.11 r = 2, n = 30, df = 1, reps = 2500 .......


Page

35


. . .
. . .

. . .


. . .
. . .

. . .



. . .
. . .

. . .







Figure Page

4.12 7 = 2, n = 30, df = 5, reps= 2500 ................. ........... 70

4.13 7 = 3, n = 30, = 0.1, reps = 1000 ................... ........ 71

4.14 7 = 3, n = 30, v 0.5, reps = 1000 ........................... 72

4.15 r = 3, n= 30, = 1, reps = 1000 .............................. 73

5.1 Umbilical Venous Blood Gas Measurements ................... .. 80

5.2 Umbilical Arterial Blood Gas Measurements .................. 81

5.3 Abdominal Arterial Blood Gas Measurements ................... 82

5.4 Physiological Measurements ................................. 83

5.5 Exercise Measurements ..................................... 84













KEY TO SYMBOLS


Symbol/Definition


a = (a,,..., as)'

lalj = j +-..- + a

A = (a,,..., at) = {aj}sxt

A' = {aj}txs

IAI

tr(A)

vec(A) = (a',...,a)'


A & B =(aB a12B


A0B=B A B)


Vector

Euclidean norm of a

Matrix

Transpose of A

Determinant of A

Trace of A

Vector of A


Direct product


Direct sum


Real numbers

Unit hypersphere of dimension p

Gradient operator

Distributed as

Independent and identically
distributed

Cumulutave distribution function


Term







Symbol/Definition


E []

Cov [., .]

V []
d

P

AN (,a2)


Expectation

Covariance

Variance

Converge in distribution

Converge in probability

Asymptotically normal r.v. with
mean I and variance a2

Chi-square r.v. with k d.f. and
noncentrality parameter A

Asymptotic relative efficiency

Term when divided by f(n) converges
to zero in probability as n -- oo

Term when divided by f(n)
is bounded in probability as n -* oo


ARE

o,(f(n))


Op(f(n"))


Term












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

A NEW NONPARAMETRIC TEST FOR INDEPENDENCE
BETWEEN TWO SETS OF VARIATES

By

Peter William Gieser

December 1993

Chairman: Ronald H. Randles
Major department: Statistics

A new nonparametric sign statistic based on interdirections is proposed for testing

whether two sets of variates are independent. This interdirection quadrant statistic

reduces to the sample coefficient of medial correlation (or quadrant statistic) when

the two sets of variates have one variable each. It has an intuitive invariance prop-

erty for the present problem and has a limiting chi-square distribution under the null

hypothesis of independence when each set of variates is elliptically symmetric, mak-

ing it asymptotically distribution-free. The new statistic is compared to the classical

normal theory competitor, Wilks' likelihood ratio criterion, and a component-wise

quadrant statistic. Using a novel model of dependence between the sets of variates

enables the computation of Pitman asymptotic relative efficiencies (ARE's). The

Pitman ARE's indicate that the interdirection quadrant statistic compares favorably

to Wilks' likelihood ratio criterion when the sets of variates have heavy-tailed dis-

tributions and is uniformly better than the component-wise quadrant statistic when

the sets of variates are spherically symmetric. A simulation study demonstrates the

relative performances of the three competitors as well as some other statistics often







found in commercial software packages. The results indicate that the interdirection

quadrant statistic performs better than the others for heavy-tailed distributions and

is competitive for distributions with moderate tail weights. Finally, several appli-

cations of the interdirection quadrant statistic are illustrated, with comparisons to

its competitors. In one example, the interdirection quadrant statistic's resistance to

outliers is demonstrated.












CHAPTER 1
INTRODUCTION


1.1 Bivariate Tests


The question of whether the pair of random variables (X, Y) are stochastically

independent, based on the random sample {(Xi, Y,), i= 1,..., n} from a continuous

distribution with density function h(x, y), has generated vast amounts of research

over the past century. After the "discovery" of the correlation concept by Galton

(1888), many bivariate measures of correlation were invented to explore the nature of

the dependency between X and Y. Examples include the classical Pearson product

moment correlation coefficient (Pearson, 1896)


r= n 1/2'
nX- X )2 ( y)2
i= i=1


and numerous rank correlation statistics based on the ranks of the Xi's (Yi's) denoted

by R1,..., R, (Qi,..., Qn) such as Spearman's rho (Spearman, 1904)
12n n R Qi ,
P n3 2 2


Kendall's tau (Greiner, 1909; Kendall, 1938)

1 '
T n= -) sgn (R R,) sgn (Q Qj),
n(n 1)i
and the sample coefficient of medial correlation, or more simply the quadrant statistic

(owing to the fact that it is based on the number of points in the four quadrants







defined by the marginal medians) (Blomqvist, 1950)
q' 1 sgn R 1 sgn Q + 1
n i=2 2


S=- sgn (, X)sgn --Y ,
Si=1

where X (Y) is the median of the Xi's (Yi's). These measures can be used to conduct

hypothesis tests for independence of X and Y. Many results are already known

about the comparison of such (suitably normalized) bivariate tests of independence,

and for various models of dependence, asymptotic relative efficiencies (ARE's) have

been computed. We present a brief summary of these results.

Farlie (1960) introduces the bivariate distribution function
H(x,y) = F(x)G(y){1 + aA(x)B(y)}, (1.1)

where F(x) and G(y) are the known marginal distribution functions of X and Y.

For (1.1) to be a bonafide cdf, d(F(x)A(x))/F(x), d(G(y)B(y))/G(y), A(x), and B(y)

need to be bounded, with A(oo) = B(oo) = 0. Clearly a is a measure of depen-

dence, with a = 0 denoting the independence of X and Y. The legal range of a is

determined by the greatest and least values of the product {d(F(x)A(x))/F(x)} x

{d(G(y)B(y))/G(y)} over all variation of x and y. Farlie (1961) then derives the

asymptotic efficiency for a generalized correlation coefficient F devised by Daniels

(1944). To each ordered pair of X's, (Xi, Xj), assign a score aij such that aij = -ai

and aii = 0. In a similar way assign scores bij using the Y's. The definition of F is

then
n n
E a]jbi
i=1 1/2


\ij=1 j= i=1 j=1 )

Gamma includes many well-known correlation coefficients as special cases, among

which are p, r, and q'. To get r for example, put aij = sgn (Ri Rj).







Farlie shows that for alternatives with the distributional form (1.1), there is always

some coefficient of Daniels' family of coefficients that is fully efficient (i.e., the Pit-

man ARE with the maximum likelihood estimator is unity). We emphasize that the

ARE's are determined by letting a 0 as n -- oo and hence are Pitman-type ARE's.

Here a is the parameter in (1.1) and not the level of the tests. Table 1.1 summa-

rizes his results when H(x, y) is the bivariate normal distribution. Note that in this

case the maximum likelihood estimator is r, and a is the "true" correlation between

X and Y. Farlie further notes that his results agree with those obtained by Blomqvist

(1950) for q' and by Stuart (1954) for p and r.


Table 1.1. Pitman ARE's Reported
by Farlie Under Bivariate Normality
ARE(p, r) 9/r2
ARE(r, r) 9/r2
ARE(q', r) 4/7r2



Konijn (1954) considers the model of dependence X = AIU + A2V and Y =

A3U + A4V, with U and V independent and AX, A2, A3, and A4 constants. The

independence case is produced by setting A2 = A3 = 0. When U and V have the

same marginal distribution G, he computes Pitman ARE's for several choices of G

by letting A2 -- 0 and A3 -- 0 as n --- oo. His results are summarized in Table 1.2.


Table 1.2. Pitman
Normal


ARE's Computed by
Uniform Parabolic


ARE(p, r) 9/72 1 0.8569 1.2656
ARE(r, r) 9/7r2 1 0.8569 1.2656
ARE(q',r) 4/7r2 1/4 0.3164 1


Konijn
Laplace







Hajek and Sidak (1967) use the model X = X* + AZ and Y = Y* + AZ
where X*, Y*, and Z are mutually independent with densities f, g, and m, re-

spectively, and A is a positive constant. They require f and g be known and that

0 < Var[Z] < oo. They focus on determining locally most powerful rank tests. When
f and g are of logistic type, the locally most powerful rank test is based on p, and

they indicate that q' is the resulting statistic when using approximate scores in the

locally most powerful rank test for f and g of double exponential type. They ac-

knowledge that they were unable to derive Pitman ARE's as A -+ 0, but Puri and

Sen (1971) successfully complete the work using a model that encompasses Hajek and

Sidak's as a special case. The general multivariate version of this model is discussed

in Section 3.3. In the bivariate case, denote the joint density function of X and Y

by ha(x,y) = f f(x Az)g(y Az) m(z)dz. Let X, Y, and Z each have unit vari-

ances and define a sequence of alternatives by An = n-1/4Ao. They show that the

Pitman ARE of the statistic T, = n-' F_,, J(F,(X.))J(Gn(Y,)) compared to r,

where J is a standardized score function and F, and Gn are the empirical cdf's of
X and Y, respectively, is given by

ARE(T,,r) = lim nA' J(F(x))J(G(y))dHA(x,y).


As an example, using the score function Jo defined as

1 if 1/2 Jo() = if u = 1/2, ,(1.2)
-1 if 0 < u< 1/2

and assuming Ha, is bivariate normal, we have that

ARE(q', r) = lim [2/7r A' sin-'(A/(1 + An))] = 4/r2.

In a series of exercises, Puri and Sen also investigate a slight variation of the Hajek

and Sidak model in the bivariate case. They use X = (1 A)X* + AZ and Y =

(1 A)Y* + AZ with X*, Y*, and Z mutually independent. Their modifications,







in conjunction with the model used by Konijn, were admittedly the inspiration for
Model 1 considered in Section 3.2.

1.2 Multivariate Tests


Our interest, however, lies in the multivariate extension of this problem. Specifi-
cally, instead of testing whether two real-valued random variables are independent, we

consider testing whether two vector-valued random variables are independent, where

the dimensions of the vectors need not be the same. Let {Xi (X ,-)X ,-X ~)', i

1,...,n} be a random sample of n pairs of vectors from a continuous distribution

with density function fx (x('), (2)), where X k) is rk x 1 and has marginal den-

sity fk ((k)), k = 1, 2 (further occurrences of the index k generally mean the state-

ment holds for both k = 1 and k = 2). Further, assume that fk ((k)) represents a

distribution that is elliptically symmetric and centered at the rk x 1 vector Ok (i.e.,

fk (a(k)) is a function of ell = (X(k) k )';k((k) Ok) alone, where ,Ek is a positive-

definite matrix.) This is a common assumption in multivariate theory as it implies
the simple structure that observations on ellipsoids defined by ell = constant are
equally likely. The elliptically symmetric class is sufficiently general to accommodate

a wide variety of different distributions, so that this assumption is not overly restric-
tive. Without loss of generality, we also take rl < r2. We are interested in testing

Ho: fx (lx), X(2)) = fi (x'))f2 (x(2)) versus Hi: XI1) and X(2) are correlated.


1.2.1 Likelihood Ratio Criterion


Wilks (1935) derived the likelihood ratio criterion for testing Ho: E12 = 0 when

the X2's are multivariate normal with mean vector p = (p', I')' and covariance

matrix E = E12 ). The normal distribution is unique, in that independence
matrxE "12 E 22







is completely determined by the form of the covariance matrix. Thus E12 = 0 is
equivalent to the null hypothesis of independence. If A = EF=I(X, X)(X, X)',

and we partition it into Aj = F"_=(X) X(i))(Xj) -. X( ))', i,j = 1,2, then the

criterion is expressed as


IA | |IA221]

= -Ir, A- A2A-A' In/2

= ITr A12A1 A12A 22

The criterion also has a convenient limiting distribution under the null hypothesis, in

that -2 l2og V2 = -nlog V r2

Muirhead (1982) shows that under the group of transformations described by

G = {g(B, c) g(B, c)(X) = BX + c}

where B = B1 P B2, (Bk nonsingular rk x rk) and c e R*+r'2, a maximal invari-

ant is the set of sample canonical correlations (Pi,..., r,) where > --- > r,

are the eigenvalues of S = A'A12AA-A'2. The usefulness of the canonical corre-

lations arises from the fact that through linear transformations on X() and X(2)

the correlation structure implicit in the covariance matrix E can be reduced to a
form involving only these parameters. In fact, the null hypothesis can be restated

as Ho: P1 = .. = Pr, = 0. Note that the nonzero eigenvalues of A12Al AA2A-2 are

identical to the eigenvalues of S, so that the criterion is (as one would expect) invari-

ant to the labeling of the two partitions of X. Since we can rewrite V as n1=,(1 p),
this shows that V is invariant under the group G. When rl is one, the lone sample

canonical correlation is just the multiple correlation coefficient between the XP1)s and

the X!2)s. Then if r2 is also one, V is just (1 -r2), the sample coefficient of alienation.
Thus, the likelihood ratio criterion is the sample vector coefficient of alienation and







a multivariate extension of the bivariate test of independence based on r. The in-

variance of V under the group 9 is an important property since it implies that a test

using V will not depend on the underlying covariance structure of either the X(')'s

or Xi2) s. Other statistics that are also functions of the eigenvalues of S (squares of

the sample canonical correlations) and hence invariant under 9, will be described in

Chapter 4.

1.2.2 Component-wise Quadrant Statistic


A nonparametric approach to the problem is explored in Puri and Sen (1971),

where a class of association parameters (and their sample counterparts) based on

component-wise ranking is defined. The statistic they propose is computed using the

elements of the matrix T = (T2 T12). The elements of Ti = IT(ij)}r are given

by
1 i(') R
T- J
$"i" n= \n+lj n+1


Here, R). is the rank of Xk) among X ,..., X() and J represents an arbitrary

(standardized) score function. Puri and Sen base their test of independence on the

statistic

SJ = IT1'
FT1 11T22 '

which is clearly analogous to V except the matrix T is used instead of the matrix A.

They also show that under the null hypothesis, -n log SJ -+ X21 2, so that their

procedure is a natural competitor of -n log V.

A score function that will be of interest to us is Jo, defined in (1.2). With this

score function, the elements of T12 are

T 2) = sgn (X X1) sgnX(2) X~2)
SI,\ s2 sS2]1
an=I







so that S'J0 is a multivariate extension of q'. A problem with statistics based on

the component-wise ranking scheme is that they do not have the desirable invariance

property exhibited by V. Thus ,S', using component-wise ranking, fails to be invari-

ant under the group g, and its performance will be influenced by the (presumably)

unknown underlying covariance structure of the X! )'s and Xv)s.

In Chapter 2, we propose a nonparametric competitor to V that like SJO is a

multivariate extension of q', but maintains the important invariance discussed in Sec-

tion 1.2. We derive the limiting null distribution in the case when 01 and 02 are

known. Sufficient conditions for asserting that the limiting distribution is unchanged

when 01 and 02 are replaced by the estimates 06 and 02 are given as well. In Chap-

ter 3, we introduce a new model of dependence, with computations leading to the

limiting distributions of the competing statistics under a contiguous sequence of al-

ternatives based on this model. Pitman ARE's are calculated using these limiting

results. In Chapter 4 we present a Monte Carlo study that corroborates empirically

the results of Chapter 3, and in Chapter 5, we apply the new test and its competi-

tors to some real-world data. We conclude with some general comments and indicate

potential areas of future research in Chapter 6.












CHAPTER 2
INTERDIRECTION QUADRANT STATISTIC


2.1 Definition


Let 0i based on X('),...,XX') and 02 based on X2),..., X") be equivariant

(under the group G) estimators of 0O and 02, respectively, such that both (0b 01)
and (02 02) are Op(n-1/2). The interdirection quadrant statistic is defined as

Qn(01, 02) = Ti2 Ecos(lrPl(X!l), Xl); 01)) cos(i2(X2), X 2);)),
i=1 =1

where Ak(Xk), X )k); k) is the interdirection proportion, first defined by Randles

(1989), between (Xk) 7k) and (X k) Yk). We now describe how to calculate this

interdirection proportion. If we let Zk) = X!k) Yk, then the number of hyperplanes

defined by the origin and rk 1 other Z(k)s (not Zk) or Zk)) such that Z k) and Z(k)
are on opposite sides, is called the interdirection count. To get the interdirection
proportion, divide by the total number of hyperplanes considered. Note that this
is the simpler and more natural divisor given by Randles. He makes a small sample
adjustment in this divisor so that his interdirection sign test is equivalent to Blumen's
test in the bivariate setting. The interdirection count measures the angular distance

between Z k) and Z(k) relative to the origin and the position of the other Z(k)s.

Randles showed that under a wide class of population models (distributions with

elliptical directions, a superset of elliptically symmetric distributions), when the X(k)s
are centered on the symmetry point of the distribution, 0k, the interdirection count
(i) is invariant under nonsingular linear transformations, (ii) uses only the direction of







each (X(k) Ok) from the origin, and (iii) has a distribution-free property. Properties

(i) and (ii) continue to hold when centering about 0k, because of its equivariance

under g but property (iii) fails to hold. Then, since the interdirection proportions

are invariant under the group g, QL(0, ,02) is as well. Notice that if r7 = r2 = 1 (the

bivariate case), then

cos(prk(X XXk)j =k)) sgn (Xk) k) sgn (Xk) k) ,


so that

Q,(O, = 10) sgn (X ( ') ,) sgn(x2) 2 ,
n Li=


which implies that Q,(X(,X(2)) = (v/nq')2. Thus, as the name indicates, Qn(l,0( 2)

is an extension of the bivariate quadrant statistic. In the next section, we derive the

limiting null distribution for the interdirection quadrant statistic in the case where

01 and 02 are assumed known.

2.2 Null Distribution When (01,.0) Is Known


In establishing the limiting null distribution of Qn(01, 02) we first seek the lim-

iting null distribution of a simpler approximating quantity. The simplicity derives

from the fact that the interdirection function pk(', '; Ok) is replaced by its expected

value pk(', "; Ok) under the null hypothesis. All calculations in this section are assumed

to be done under the null hypothesis, so no further explicit mention of this fact will

be made. Because Q(01,02) is invariant under the group 9, without loss of generality

we assume that fk (X(k)) represents a distribution that is spherically symmetric and

centered at the origin (i.e., Ok = 0 and Ek = Irk.) Define
n n
Qn = r r 2 EE cos(7r(i, )) cs(rp2(i, j)),
n i=1 j=l







where we have kept just the subscripts i and j, since the subscript k on pk(i,j) (and

also pk(i,j)) indicates whether it is a function of the X(')'s or X(2)'s, and

k(i, j) = E,, [p)k(i, ) I Xk), X3k)

a hyperplane defined by r'k 1
= P X(k)'s and the origin lies between X(k), Xk)
Xk) and X(k)

= (Angle between X k) and Xk))/lr.


The last equality follows because of the underlying spherical symmetry. Let X(k)

R(k)U(k), where R(k) = IIX(k)ll and U(k) = X(k)1/1X(k)ll. Spherical symmetry guar-

antees that Rk),.. ., Rk) are positive quantities independent of Uk),..., U~), and

the U k)s are iid uniform on the unit hypersphere of dimension rk. Here U(k) is the

direction of X(), and it is easy to see that both Pk(i,j) and pk(i,j) depend only on

the directions U k) and U.k). Now

cos(7rpk(i,j))= cos(angle between U0k) and Uk)) = U(k)'k)

so that
n 7'17'2 1) 1) 2)'U (2)
Si=1 j=1
(2.1)
1 ri r2 n ( 2)
Ss=l 1 t= =1l


Some moments involving U1') and U(2) will be useful in subsequent derivations. If

U1') and U(2) are independent with U(') Uniform(fr,) and U(2) Uniform(rn,),







then it is easy to show that

EH [U(k) = 0


EH0 [U(k)U(k)'] = Ik
7'k
(2.2)
EH [U(k)'MU(k)] = vec(M)'vec(2.2)


EH [U)'MiU(2)()'M (2) = --vec (M)'vec (M2).


Two basic classical limit results are restated here for future reference.

Lemma 2.2.1 a'Z, ~ AN(a'p., a'Ea) for any a f 0 if and only if Z, ~ AN(p, E).

Proof of Lemma 2.2.1 (See Serfling, 1980, p. 18). O


Lemma 2.2.2 If Z,,. AN(t, Ik), then Z' Z,n -d x('l).

Proof of Lemma 2.2.2 (See Serfling, 1980, p. 128). O

We now have all the necessary tools to derive the asymptotic distribution of the

interdirection quadrant statistic. We begin by finding the limiting distribution of the

approximating quantity Q,.

Theorem 2.2.1 Q, d Xrl2


Proof of Theorem 2.2.1 Let B = {bst}Tr xr be an arbitrary matrix of constants that

are not all zero and define Z = {E= --rU,(U(2)Ul},,. Then
1 2 ta
vec(B)'vec(Z)= E :b*tZ,.
s=1 t=1


a=1 2=1 t=1)


CV=1






which, using (2.2), is seen to be a sum of iid random variables with mean zero and vari-
ance vec(B)'vec(B). Now n-I1vec(B)'vec(Z) ~ AN(O,vec(B)'vec(B)) via the

central limit theorem. Lemmas 2.2.1 and 2.2.2 then imply that n-'vec (Z)' vec (Z) -d

Xrr2. The result follows by noting that n-'vec(Z)'vec(Z) = n-'1 EL E= Zjt,
which, referring to (2.1), is just Q,,. 0

Now using arguments similar to those in Randles (1989), we are able to find the
limiting distribution of Q,,.

Theorem 2.2.2 Qn, Xr1.

Proof of Theorem 2.2.2

EH, [(Qn 2Qn )2


E= n lr2 cos(7rfl(i,j))cos(,rP2(i,j)) cos(rpi(i,j))cos(7rp2(i,j))}

2 2 n i
n2t2 Y Ewo{ cos(7rPi(i'j)) cos(rp2(i'J) cos(Trpi(i,j)) cos(p2(ij))


x { cos(7rp(i',j')) cos(ir2(i',j')) cos(7rp(i',j')) cos(7rp2(i',j'))}

2r2 r 2n(n 1) 12]
= 2rrn(n EH, [f cos(r (i, j)) cos(72(i, j)) cos(7rp(i, j)) cos(rp2(i,j))} ,


where the last equality is seen by considering the following. Let U1) = DiAi, where

Di = sgn (Uf1)) and A, = sgn (Up1') U '). Then A, shows the observed axis and Di
indicates which end of the axis was observed. Note that Di is independent of Ai and
D1,..., D, are iid Bernoulli random variables with probability of success 1/2. Taking
the expectation first with respect to the D's, when one or more of the four subscripts
is unique, the expected value is zero. Further, if i = j or i' = j' the integrand is
zero since pk(i, i) = pk(i, i) = 0. Now the last expectation converges to zero because







Randles has shown that k(i,j) = Pk (i,j)+op(l), and the integrand is bounded so the

Lebesgue Dominated Convergence Theorem can be applied. Thus, Q = Q, + op(l),
and Theorem 2.2.1 yields the desired result. O

We have now established that Q,, has a convenient asymptotic null distribution,
which makes it a viable option for the present hypothesis testing scenario. Further,
since it has the identical limiting null distribution as -n log V and -n log S'J, the
relative performance of the three competitors can be fairly measured. However, in
practice the values of 01 and 02 are rarely known so that they must be estimated.
The next section considers this situation.

2.3 Null Distribution When (010.9) Is Unkniiown


Unfortunately, when (06,02) is replaced by (61,02), the proof of convergence to
the ),'2. distribution is much more difficult. We wish to find sufficient conditions

under which Q,(O61,02) = Q,(0,02) + op(i). Since

Q.(01,02)- Qn{81,02)

I= ,. {cos(7rpi (i, j; 0i))cos(7rP2(i, j; 02)) cos(7rPi(i,j; 01))cos(7rP2(i,j; 02))}


2 {cOs(7rp (i',j; 01)) cos(rp2(i,j; 62))- cos(7Tpl(i, j; 0l)) cos(ip2(i, j; 2))}
r1,1

+ r_2 {cos(7rpi,(i,j; 6i)) cos(7rp2(i, j; 02)) cos(7rp(i,j; 0I)) cos(rP2(ij; 02))}


= B1, + B2n, (2.3)


it suffices to show that both Bl, 0 and B2, 0. Consider B1, first. The strategy
for dealing with B1, is to show that it suffices for the second conditional moment
of B1,, to converge in probability to zero. The conditional moment is useful in that







it "separates" Bi, into a sum of terms whose factors involve only the X(1)'s or only
the X(2)'s. The limiting behavior of these factors is then established in Appendix A.
More formally, let A, = {X1), i = 1,...,n}, ,X2 = {X ), i = 1,...,n}, and R =
{Ri, i= 1,..., n}, where R is a permutation of the integers {l,..., n}. Conditional
on X1 and X2, the only random component of B1, is the way in which the X(')'s
and X(2)'s are matched. In other words, given X1 and X2, under the null hypothesis

B,, is a function of {(X(),X 2)), i = 1,...,n}, where R is uniformly distributed
over all permutations of {1,..., n}. Given this setup, we prove the following lemma.

Lemma 2.3.1 If EH0 [B X1, XX2] 0, then B1 -P 0.

Proof of Lemma 2.3.1 Since

Bin = -12 COs(7rp1(z,J; i)) {cOs(7rP2(,j; 2)) cos(7rP2(,j; 2))},



G,(X1, X2) EHo [BE I XX, 2] = EHo [ r2 c(Ri,R)d() ]


where
c(i,j) = cos(7rPni(i,j; 0))

and
d(i,j) = cos(rP2'(i,j; 62)) cos(7rP2(i,j; 02)).

For > 0 and > 0,

P [IBli > e] = P IBii > e, Gn(X,,X2) > ]+ P [B. > E, Gn(Xl,X2) < 2


2 + EH [I (IB > e, Gn(X1, X2) < 2







which for n sufficiently large,


S, + EHo


&
S+ EHo





= + EHo

< +EH6
S2 Ho


62
< 2 P [IBu, > 1 X21

f 26 1 B 2 X,| v1]
)2 '-2E [ In


< 2 iG.(XI, X2)


=6. O


In light of Lemma 2.3.1, it suffices to show G,(X1,X2) -4* 0. With this in mind, define

S= rr2 c(Ri, Rj)d(i,j) = rr Ec(Ri, Rj)d(i,j)
n in in j


where the last equality follows since d(i,i) = 0. This means that G,(X,,X2)

EHo [S2], so that in order to write out the expression for G,(X1,X2) we need only
find the second moment of S. First, we will need some preliminary results regard-
ing the moments of c(Ri, Rj). Let Pn represent the collection of all permutations

of {1,...,n}, ,Ck the collection of all subsets of size k from {1,...,n}, and ,Dk the
collection of all subsets of size k and their permutations from {1,..., n}.


I (G,(X, X)



[I G, (x, ,2)




Sl Gn(X2
E2 2 1


EHo I fll| > 2, Gn( ',) < 2 X







Clearly, for i Z j,

EHo [c(Ri, R)]= c(a, a,)P [ =a]
ae Pn


= c(f, #2)P Ri =i,j = 2
/3E nD2


- I C(01, 2)
n(n 1) D2
PE nD2


S

and in a completely analogous way we have for i $ j and i' $ j' that if i' = i


and j' = j, then

COVHo [c(R, Rj), (Ri,,Rj,)] = VHo [c(i,j)]


=n E { c(#l, #2) -ij
n(n- 1)
/3 nD2
Cl
n(n- 1)'

and if i' = i or j' = j (but not both), then

Covo [c(R, Rj)c(R,, Rj,)i = COVHo [c(Ri, Rj)c(Ri, Rj,)]
an fi 1o '=J btntbohte


1
n(n 1)(n 2)
P3E D3


{c(1, #2) } {c(31,/ 3) -


C2
n(n 1)(n- 2)'

and if neither i' = i nor j' = j, then

CovHo [c(Ri, Rj)c(Ri, Rj,)]


1
n(n l)(n- 2)(n 3)
PE ED4
C3
n(n l)(n 2)(n 3)


{c(I, 12) c}{c(/3, 4) -







Let d, dl, d2, and d3 be the analogous quantities in d(i,j). Now using these expressions

for COVHo [c(Ri, Rj)c(R, R')], we see that

VHo [S] = VHo d(i, j)c(Ri, R)
.A'j)


= Z d(i,j)d(i',j')CovHo


__Cl
n(n 1)
P~nl)Dz


4c2
d(n( 1)(n 2)
n(n 1)(n 2)


+ c d(I, 02)d(#3, #4),
n(n 1)(n 2)(n 3) 3E (4D4
/3E D


and since


,1R [V
VHo j d(i, j)c(Ri, R) = VHo EI{d(i,j) d}c(Ri, R) ,
i J kii


we have that
S cdl 4c2d2 C3d3
n(n 1) n(n l)(n 2) n(n 1)(n -- 2)(n 3)

Note also that since En {c(i,j) c} = 0,


- E = cl + 4c2 + C3 = 0,


S{c(i,j)


so that c3 = -(c1 + 4c2), where a similar result holds for the d's. Finally,

EHo [S] = EHo Ed(i,j)c(R, R,)


= d(i, j)EHo [c(Ri, Rj)


= c d(i,j)
i~j


= n(n 1)cd.


E D
/6e Da


d(0A, 02)d(#,, #3)


[c(Ri, R,)c(R', R)]






Since EHo [S2] = VHo [S] + E2o [S], G,(X1, X2) can be expressed as

.2 2 c dl 4c2d2
n2 n(n- 1) n(n 1)(n 2)

(ci + 4c2)(d + 4d2) 2
n(n- 1)(n 2)(n 3) + ( -)


To show G,(X, X2) -p 0 (and hence that Bin 4 0), it suffices to show c = op(n-/2)
cl = O(n2), c2 = Op(n5/2), d = Op(n-1/2), dl = op(n2), and d2 = op(n5/2). As stated,
these results, along with the requisite assumptions needed, are in Appendix A.
Recalling that Q,(Bl, 2)- Qn(01, 02) = Bl + B2n (see (2.3)), we must now show

B2n -2 0 to complete the argument that Q 2(1, 2) = Q,(0, 02) + Op(l). Of course
it suffices to show EHo [Bj] -2 0. Since

B2 = r2 cos(r 2(i,j; 02)) {cos(r(i,j; 0)) cos(7r1(i,j; 01))},
B n E


we have
22 n
EHo [BLn r r2 EHo [cos(7rp2(i,j; 02)) cos(7rJ (i,'; 02))]


x EHo [{cos(7rp(i, ; 1i)) cos(7rP(i,j; 0i))}

x {cos(rpl (i', j; 0b)) cos(7rl (iZ', j; 1))]

2r r22n(n 1) EH
= 2r r --E H [cos2(7i2(,j; 02))]

x EHo [{cOs(7 (Zi,J; )) COs(7r1(i, ; 02))} 2

where the last statement follows from logic similar to that used in Theorem 2.2.2.
Then using the fact that the integrand in the second expectation is bounded, applying
Lemma A.0.6 and the Lebesgue Dominated Convergence Theorem yields the result.













CHAPTER 3
PITMAN ASYMPTOTIC RELATIVE EFFICIENCIES


3.1 Introduction


To compute Pitman ARE's, a model of dependence must be adopted to serve as an

alternative to the null hypothesis of independence. Konijn (1954, p. 300) states that

"the crucial point is the specification of a class of alternatives which is (i) sufficiently

wide to include some approximation to any situation that may arise in this class

of problems, and (ii) manageable mathematically." For tests involving a change in

location of a distribution, shift alternatives form a satisfactory idealization to a wide

class of problems and are quite amenable to mathematical analysis. But because of the

innumerable ways dependence can manifest itself, our situation cannot be expected to

lend itself as easily to so simple a model. This is not necessarily a fatal blow, however,

since when considering a model in the context of local alternatives, there is reason to

believe the specific form of the model is of little consequence. Witness the agreement

in the Pitman ARE's reported in Section 1.1 for the bivariate case when several

different models were used. Thus, although we propose a model that is intuitively

appealing in some aspects, our main reason for choosing it is for its mathematical

tractability. We require that the model be a function of a nonnegative real-valued

parameter A such that as A -+ 0, the sequence of alternatives defined by this model

will converge to the null hypothesis. In fact, it is necessary that the convergence

of this sequence of alternatives occurs at such a rate so that it is contiguous to the

null hypothesis. This necessity is two-fold. In doing calculations under the sequence

of alternatives, contiguity allows us to use an approximating quantity in finding the






limiting distribution of statistics of interest and also aids in determining the form of
that limiting distribution.

3.2 Model 1


A generalization of the model apparently first studied by Konijn (1954) is given
by

(x(1) ((1A)Y(1) + AM, Y(2)
S X() AM2Y(') + (1 A)Y(2)

(1 A)I,1 AMr YM
AM2 (1 A)Ij y(2)


= AA y(2) = AAY,


where Y(') and y(2) are independent random vectors that are rl x 1 and r2 x 1, respec-
tively, M1 and M2 are arbitrary (known) matrices of dimensions ri x r2 and r2 x rl,
respectively, and 0 < A < 1/2. Notice that for rl = r2 =- r and M1 = M2 = I,, A =
1/2 implies X1') = X(2) (perfect correlation), while A = 0 corresponds to the null
hypothesis of independence. Thus we can restate the testing problem as Ho : A = 0
vs. H1 : 0 < A < 1/2. Since Y = A'1X is a nonsingular linear transformation,
the density function of X can be expressed as fx (x; A) = abs (IAA-1) fy (Ax'z),

where fy (y) = ft (Y('))f2 (y(2)) is the density function of Y. We assume that the

distributions of both Y(') and y(2) are elliptically symmetric with dispersion param-
eters E, and E2, respectively, and are centered at 81 and 02, respectively. In other
words, fk ((k)) = Ck9k (((k) Ok)' (Z(k) Ok)), where Ek is a positive-definite
matrix and gk () does not depend on Ok or Ek.







Such a model might conceivably arise when considering a battery of psychological

or psychophysical tests administered to a group of subjects, with the goal of classifying

the outcomes relative to certain independent "factors." Suppose that apparently the

outcomes of one set of tests are practically determined by one factor, and the outcomes

of another set of tests are practically determined by a second independent factor. In

order to test this hypothesis, Model 1 could be used, since the alternative might

reasonably be that all the outcomes depend, to varying degrees, on both factors.

3.2.1 Contiguity


We wish to show that the sequence of alternatives H1 : An = n-1/2Ao, where

Ao > 0, is contiguous to the null hypothesis. To achieve this we follow the rationale

of Hajek and Sidak (1967, pp. 201-214), which we outline here. Let L(; A,,) =

fx (x; An)/fx (x; 0), and An = log H=i, L(Xi; An) = E?1 log L(Xi; A,). LeCam's
first lemma asserts that if A,, AN(-o,2/2, a2), then the densities Hn=1 fx ('i; A,)

are contiguous to the densities nf=1 fx (ai; 0). Another way of expressing this is to

say that the sequence of alternatives A, is contiguous to the null hypothesis (Ao = 0).

If W,, = 2 X =1[L(Xi; An)/2 1], LeCam's second lemma states that contiguity will

follow if, under Ho, the summands log L(Xi; A,) are uniformly asymptotically negli-

gible (UAN) and Wn AN(-a2/4, a2). Because the summands depend on n, finding

the limiting distribution of Wn directly is quite difficult so they consider the first-order

approximation of W,. This approximation is expressed as T, = A, E'= L'(Xi; 0),

where L'(Z ; 0) L(x; A) l=0. Hajek and Sidik demonstrate the contiguity for

a univariate shift alternative (pp. 210-213) and a univariate scale alternative (pp.

213-214) by showing that under the null hypothesis, W, = Tn a2/4 + o,(l), the

UAN condition holds, and T,, AN(O, a2). Randles (1989) has extended the argu-

ments to show contiguity for a multivariate shift alternative. Noting that Model 1






is a multivariate extension of a scale alternative leads us to emulate the methods of
Randles in extending the proof of contiguity. A sketch of this extension is included in
Appendix B. Since we will need the form of T,, in determining limiting distributions
under A,, we prove the asymptotic normality of T,, presently. Considering previ-
ous discussion regarding the invariance of the statistics -n log V and Q,n, we assume
hereafter that Ok = 0 and Ek = Irk. We discuss the ramifications of this assumption
with respect to -n log SJO later. First we need to find the expression for L'(a; 0).

Lemma 3.2.1 For Model 1 (given in (3.1)),

L'(x;0) = 2 (x()'x(1)1()x)'(1)) +)

+2 X(2)X(2)02(x(2)'a(2)) +r2)

2x(1 (O(X(1)'1(1))M, + 2((x'2 (2)M) M (2)

where ck(t) = gk' (t)/gk (t).

Proof of Lemma 3.2.1 Since

L(; A) =abs (IAAlI) fY(AZx,
fy ()

we see that

L'(x; A) = abs (JAA-1) + abs (AA s ') a fY- (A'z
A \ fy (x) WA fy (X)

Now

-abs (IAAi-') = -abs (JAA-1) tr (-AtP),

where

9 I,, -Mi)
P -= AA =
A \-M2 I2








abs (AA )
A=0


= abs (IAol-1) tr (Ao'P)


=abs (I,, r-) tr (I:2P)

= tr(P) = 7i + r2.


Also


a fy (AAx)
iA fy (x)


S A -A Vfy (x)


fy (X)
= A 'PA ^ ,


so that


fy (A'1x)
fy (x)


S(A-' PA-,x), Vfy *()
SPfy ()


I PI/()2


= (P()
= (L zv) ())
-f^ W


{ I,, M, 1)
-M2 Ir2, )


= 2(x(') M, x(2))'X(1)1(x(1)'x(1))

+ 2(-M2X(') + x(2))',(2) 2(X(2)'x(2))


because


Vfk (k)) ) ((k) (k)
fk (Xk) ( k)'(k) '


The result follows immediately. 0


so that






Recall that we can represent X(k) as R(k)U(k) when X(k) has a spherically symmetric
distribution (see Section 2.2). Using this form for X() gives

12 { ( )201 ))2) +"I
T. = n-V 2nAo R M(') (( + -2


+ ((R2)2 2((R 2)2) + U1)RU2)

where
R, = R)R 2 ( ((RM'))2)M, + ((2))2)M)

Lemma 3.2.2 If EHo [(R(k))4k~((R(k)2)] < 00, EHo [(R(k))202((R(k))2)] < 0o, and

EH0 [(R(k))2] < oo, then VHo [T,] = a2 < oo and T, ~ AN(0, a2).

Proof of Lemma 3.2.2 To guarantee that a2 < oo, it is sufficient for

EHo [{(R))21 ((R1)2) + ) + ((R2)2 2(( 2))2) + U(1)'RU(2) <
2 2 )

where
R = R()R(2) ((1 ((1))2)Ml + (( (2))2)M2)


Thus ,2 < oo if EH0 [(R(k))402((R(k))] < Co and EH [(1)'RU(2)2] < where

the second expectation is easily seen to be finite if EHo [(R(k))22 (k))2) < oo and

EHo [(R(k))2] < oo. Appendix B shows further that if o2 < oo, then EH0 [T,] = 0.
Since the terms are iid, an application of the central limit theorem gives the result.



Thus, we have established conditions under which contiguity holds. In the next
section we work out the limiting distributions of Q,, -n log V, and -n log SJO.






3.2.2 Limiting Distributions Under A,,


LeCam's third lemma states that if, under Ho, () ~ AN((_',/), ( 12
t ) 612 U12))
then S, ~ AN(1( + a012, o) under a contiguous sequence of alternatives. Find-

ing the limiting distributions of Q,, -n log V, and -n log S'O under the the con-

tiguous sequence of alternatives A,, will involve showing that under Ho, ()~

AN((), (7 1))' where S, is an appropriately defined statistic. Then, under A,,
S,' ~ AN(ol,2, Or) and from this it will be possible to determine the limiting distri-
bution of the statistic of interest. We begin by finding the asymptotic distribution

for Q,. We assume throughout the rest of the chapter that the moment conditions
in Lemma 3.2.2 are satisfied.

Theorem 3.2.1 Under A,,

Qn Xr, (4 ver (Eo [R])'vec (EHo [R])) .


Proof of Theorem 3.2.1 Let a = (a,, a2)' be an arbitrary pair of constants not both

zero. If S,, = n-' /2 U!1'(v1 2B)U (cf. Theorem 2.2.1), then





n-1/2 ^ 2oa ((1)()2R1 ))2) + a) +a ((R 2))202((R 2))2) + r2)
2 o a2 22) 2 (2) 2


+ U 2)' +.2 i + 2)r2A B






Since EHo [5',] and EHo [T,] are zero, EHo [a'(S,, Tn)'] = 0. Also, the summands are iid
where the three terms in each summand are uncorrelated with mean zero so that,
VHo [a'(,5, T,)] = afVHo [,S] + a"VHo [Tn] + 2ala2CovHo [S, Tn]

= alvec (B)'vec (B) + a'r2

+ 2al a2 2Ao iE2EH [U()'RU(2)U()'BU(2)] .

Further, using (2.2),

EHo [U(1)'RU(2)U(1)'BU(2) = EHo [U(1)'RB'U()]


S- vec(B)'vec(EH [R]),


so that of = vec (B)' vec (B), a2 = or2, and r12 = vec(B)' ( ^--vec (EHo [R])). The

asymptotic normality of a'(Sn, Tn)' follows by applying the central limit theorem.
Then under A,,

Sn ~ AN vec (B) -----vec(EH [R]) vec(B)'vec(B)


with Lemmas 2.2.1 and 2.2.2 giving

Qn 4A vec2 (EH [R)' vec (EHO [R)) .


The result follows by noting that since the difference between Qn and Qn converges in
probability to zero under the null hypothesis (see the proof of Theorem 2.2.2), by def-
inition, the same difference will converge in probability to zero under any contiguous

sequence of alternatives. Thus Qn and Qn will have the same limiting distribution
under A,. O

We next find the asymptotic distribution of -n log V under An. It is easy to

show, using U-statistics, that if EHo [X'] < oo, then -n log V Xrr2 under Ho. To







determine the limiting distribution under A,,, we need to find a simple approximating

quantity. Puri and Sen (1971, p. 364) show that
rI rl1 r2 r2
-n log V In ss i' ( ,i)'i' = Op(n-1)
s=1 s'=1 t=1 t'=1


where p, the sample correlation matrix, is partitioned into p =- {^ss'}'ri),, ^

{ P,}r2xr2, and Pi2 = {(st}rixr2 = P,. Since Pkk = Ir, + Op(1), we use Slutsky's
Theorem to achieve the following simplification,

-nlog V = nvec (P12)' (P1 0 P21)vec (p2) + Op(n-1)

= nvec (Pi2)' vec (i2) + -(1).


Then, since P12 = n-' A2 and n-' op1 X) = op(1), we use Slutsky's theorem again

to get

-n log V = n-'vec (A12)' vec (A12) + p(1)


= n-vec X ')X, 2 vec X)X 2)+ o()
i=l i=l

= -'I tr (( X 2)')'X ) + ,(1)
i=1 =l
(3.2)
n n
= n- X )'X( ')X(2)'(2)+ o (1)
i=1 j=1

S O) R ) R (R2) R 2) (1) (1) (2) 2) (1).
i=1 j=1


Thus we have a convenient approximating quantity for -n log V. In fact, comparing

the approximations for Q, in (2.1) and -n log V in (3.2) reveals the underlying simi-

larity in structure between these two statistics, which at first is not readily apparent.

To find the limiting distribution of -n log V under A,, we proceed as in the proof of

Theorem 3.2.1.







Theorem 3.2.2 Under A,,,

-nlogV X2~ (ovec(EH [R()R(2R) ec(EH [R()R)R) .

Pruof of Tliera-nm 3.2.2 Let

S, =- n1'/vec (B)' vec X} ')X 2)


Sn-1/2 X')BX!2)


= n-'/2 U!1)'(R 1) 2B) 2)
i=1

The key quantity is again
1-2 = CovHo [Sn, Tn]

= 2AoEHo [U()'RU(2)'U()R(1)R(2)BU (2)]

= vec(B)' ( vec (EH0 [R R(2)RR)).

Applying Lemma 2.2.1 and then Lemma 2.2.2 yields the result. E

If we adopt the notations sgn (a) = (sgn (xil),..., sgn (x,))' and i = (1,..., ,n)',
then use of the score function Jo defined by (1.2) enables us to represent the matrix T12
described in Section 1.2.2 as
T12 = n-' sgn (X )- )) sgn (X(X X'(2)'.
i=1

Further, using Theorem 2.13 in Randles (1982) on the components of T12, we are free
to replace the sample medians by the population medians (which we have assumed
are zero) when considering the asymptotic distribution of T12. Continuing, we now
have
T1 = n-1 sgn (X')) sgn (X2))' + o,(1)
i=1

= n- Csgn (U')) sgn (U2)) + O(l).
i=







Combining this with the result that -n log Sa = nvec (T,2)'vec (T,2) + Op(n-') (see
Puri & Sen, 1971, p. 359), shows that

7 log SJo


= n-1vec sgn (U 1) sgn (U )vec sgn (U') sgn (U2)' + o (l)
= 71P\i=lkl


= n-1 ii gn (U)sgn ) sgn (U))' sgn (U2)) + o,(l),
i=1 j=I


where again it is of interest to note how this approximation compares to those of Q,
in (2.1) and -n logV in (3.2).

On a cautionary note, because of the noninvariance of -n log S'J, all subsequent

derivations apply only when Ek is in reality a diagonal matrix. Recall with the other
statistics there was no loss of generality because of their invariance under g. This is
probably not a serious concern, since as Randles (1989) has noted in the comparison of
his (affine-invariant) interdirection sign test with a (nonaffine-invariant) component-
wise sign test, spherical symmetry is favorable to the component-wise test. He further
states that although it might be possible to improve slightly the efficiency of the
component-wise test over certain points in the alternative, the result is a drastic
depreciation of its efficiency over the rest of the alternative space. This generally is
not a desirable property of a statistic. The same logic applies in the present situation.
After deriving a moment needed in the subsequent theorem, we proceed exactly as in
Theorem 3.2.1.

Lemma 3.2.3 If U(k) Uniform(Q,k), then

Er[I ]
[IU(k)l] 2
1 2







Proof of Lemma 3.2.3 A point on an rk-dimensional unit sphere can be uniquely rep-

resented (provided rk > 2) by 7rk 1 angles 71,... rk_1 and the equations

1U/) = sin 71 sin 72 ... sin r,,-2 sin 7q,_k

Uk = sin 711 sin 772 ...sin 77k-2 COS 77r-1

((k) = sin i sin 772 ... sin 7k-3 cos 77r-2

k) = sin 791 sin 772 .. sin r/7-4 COS 7,r-3




Uk 2 = sin 7 sin 72 COS 773

Urk1 = sin 71 cos 72

1(k) = COS 71.


If the joint density of the angles is proportional to sin'k-2 771 sin5k-3 772 *.. sin 77r-2,

where 0 < j7i < nr, i = 1,...,rk 2 and 0 < 7rk-1 < 27r, then U(k) will have the

uniform distribution on the unit hypersphere of dimension rk. Clearly the angles are

independent, with 77rk-l uniformly distributed on (0, 2r) and the other angles having

power sine densities on (0, r). For instance, the marginal density for 771 is


g(71) = 2 sinrk2 7I1.
k-1
V 7- r( 2 )







Thus,


E [Ul = gI cos ig(,)d,
<- J J


cos 77 sin'"-2 71- d17I


21' (k 1
/-02l~V


v()
S(2 zrk-2dz
7'k -- 10


2r (2







Since IU(k)I = 1 when rk = 1, the formula holds for rk > 1. By symmetry, the result
holds for E [jUIk)l]. O

Theorem 3.2.3 Under A,


-n log SJ X172


Proof of Theorem 3.2.3 Let S, = n-1'2 1 i/ l sgn (U1))' Bsgn (U02)) so that the key
quantity is again
012 = COVH [Sn, Tn


= 2AoEHo [()' U()sgn (U')I Bsgn (U(2))]

= vec(B)' (2AoEH [/1()l] EHo [IU(2)1] vec (EHo [RI))


(EHo [R])'vec(EHo [R]) .






and using Lemma 3.2.3,


= vec(B)' r2AoF (2 r2) "EH [R])
r 12 r( 2(2EH

with Lemma 2.2.1 and Lemma 2.2.2 giving the result. [

3.2.3 Comparison of Statistics


We are now in a position to find the expressions for ARE(QU, -n log V) and
ARE(Qn, -n log S'J). Since each of the statistics have limiting noncentral x 12 dis-
tributions under A,, Hannan (1956) has shown that the Pitman ARE is the ratio of
the noncentrality parameters. Referring to Theorem 3.2.1 and Theorem 3.2.2 we are
able to report that
r r2vec (EHo [R]) vec (EHo [R])
ARE(Q,, -n log V) = R
vec (EHo [R(1)R(2)R]) vec (EHo [R()R(2)R])

4vec(plM1i + yo2M')'vec (lM, + 2M':)
r r2vec (Mi + M2)'vec(Mi + M'))

where
1 = EHo [R 2)] EHo [R(1)1((1)2)

and
V2 = EHo [R(1)] EHo [(2)2((R2)2)].

Note that for M, = M',

ARE(Ql -n log V) = (9' + 2)2 (3.3)
rlr2

Also, using Theorem 3.2.1 and Theorem 3.2.3 we see that

ARE(n,, -n log SJO) = 2- 2 r 2r2
2 (-2)







where it is of interest to note that there is no dependence on gi and g2 or the form

of the matrices M1 and M2. Of course, as noted earlier, this result holds only when

Ekk is diagonal. For general Ek, (i.e., elliptically symmetric distributions), the Pit-

man ARE will depend on the underlying covariance structure of the X(')'s and X(*)'s.

In the next sections, we compute ARE(Q,,, -n log V) for various choices of gi and g2.

Because the formulas for the Pitman ARE's are quite complex, a visual aid can

help reveal some of their structure. Therefore, we provide graphs which illustrate

the Pitman ARE's for the three statistics in various simplified situations. One sim-

plification is that identical distributions were used for both X(1) and X(2). Recall

that the Pitman ARE is the ratio of sample sizes needed for competing tests to

maintain the same limiting power and size when converging to the null hypothe-

sis. Thus, if T, and S, are two competing sequences of tests, then loosely speaking,

ARE(T,, S,) > (=, <)1 implies that T,, requires fewer (equal, more) observations to

maintain about the same power as S,, meaning Tn is more (equally as, less) efficient.

Since normally ARE(T,,S,) is not necessarily bounded, we have chosen to plot the

Pitman ARE's for the dimensions given by the axes labeled rl and r2 using the trans-

formation 1/(1+ARE(S,,Tn)). This results in the surface lying between 0 and 1 for

any possible value of ARE(S,,T,). Theoretically, this keeps the visual comparison

of situations where the competing statistics relative performances change direction

on equal footing. The surface is now loosely interpreted as the ratio of the sample

size required for T, to the sum of the sample sizes required for both T, and S,.

Practically, this means that in all cases T, is doing better if the surface is below 1/2

and worse if it's above 1/2. For example, in Figure 3.1, Q, is in the second position

and the surface is below 1/2 (except at rl = r2 = 1 when Q, and -n log S0 are

asymptotically equivalent), so it is more efficient than -nlog S' when at least one

of r1 or r2 is bigger than 1 and equally as efficient in the bivariate case. Similar






graphs of the Pitman ARE of Q, with -n log V follow the sections in which specific
distributions have been assumed, allowing for the evaluation of Wp and p~2.











1

0.75

0.25

0.25


4 4 5

3 3
r2 2 2 rl

1







Figure 3.1. 1/(1+ARE(-n log SJO, Qn))


Exponential power class


A convenient elliptically symmetric class of distributions is the exponential power
class. The exponential parameter v allows for a choice of distributions with varying







heaviness in their tails. This will enable us to evaluate ARE(Q,, -n log V) when the

underlying distributions of X') and X(') have either heavy or light tails. When

v = 1, this corresponds to the multivariate normal distribution. When 0 < v < 1, the
resulting distribution has heavier tails than the normal and when v > 1, the resulting

distribution has lighter tails than the normal. In fact, as v -- oo, the distribution
becomes uniform.
Let

9k (t) =exp -- ,
\ Ck

and

(r k 'fk + 2 k/2 'k
Gk k_ 2 2V ck 2vk =
r(rk T) Lrr Tkk + 2)j k
2Vk 2Vk 2Vk

Then X(k) has the exponential power distribution located at the origin with dispersion

parameter ITk and exponential parameter Vk (X(k) ~ Exp(vk)). The density function

of (R(k))2 is given by:

hkr(t) k/2 2-1 xp -(- -t" (3.4)



We first calculate a useful moment.

Lemma 3.2.4


EHo [(R(k ))l k d/2 provided rk + a > 0.
r (4" )







Proof of Lemma 3.2.4 Let T have the density h given by (3.4). Then

CkTr"&/2 00
E [T t(k/2 (rk+)/2-l exp(-(t/dk)k)d
2

= kK.2d+)/ t(rk+a)/(2vk)-1 exp(-t)dt

ik

k.+o>/,, ( ,2'+_)/2+ ak
_r"k\ 2(ck ) 2vk


(r (_+ a)
\ 2vk ) a/2
(r r ) k
2Vk/

so long as rk + a > 0. O

Theorem 3.2.4 If X(k) ~ Exp(vk), with Vk > -(rk 2)/4, and we assume that
M1 = M', then

ARE(Q, -n log V; V1, V2)

-1/ + I2 r ( r + 2v, 1



d )21/22 r2
d dl ^ /2 2V2 2V12y1
= -V



+ V2 (dd 2) ( r2 ) r, ( r
2v, 2v2



Proof of Theorem 3.2.4 Since

k(t) = gk (t)/gk (t)= k -1
Ck









V- EH IR(' EHo [( R)R )2(]

=-- EHo [RI( EHo [R
c1

r2 (2 1 ( r + 2v1 12_
V1 \2v d/2 2i' d(2vI/i-1)/2
-d2/2 dp(1

1 2 ( 2+1 r_+2v) -1

-VId212 r2
(dl, r r2 )r ( i1
2v2 2v,

and similarly,
(r, + (r2 + 2v2 1
Q) r ) r( 2)r2r2


The moment conditions in Lemma 3.2.2 are easily shown to be satisfied for vL >
-(rk 2)/4, so putting these values of Vp and ''2 in (3.3) gives the result. O

Note that the condition that Vk > -(rk-2)/4 is really a restriction only when rk = 1,
in which case we need Vk > 1/4.

Corollary 3.2.1 If va = v2 = v and r71 = r2 then
2Fr r+ r+2v-1 2
ARE(Q,, -n log V; v; r7) = 2( v 2r 1) (3.5)
-r

It is interesting to compare this expression with the Pitman ARE of Randles' interdi-
rection sign statistic, V,,, and Hotelling's T2 from the multivariate location problem.
It turns out that
.2 r+l)
ARE(Q, -n log V; v; r) = 2 2 ARE(V,,2;;.
S)2vj )







Inserting specific values for r and v in (3.5), we have that


ARE(Q,,, -n log V; v; 1) = --V -
2v -
2 1v


ARE(Q,, -n log V; 1; r-) ,



and

ARE(Q,, -n log V; 0.5; r) = 1,

where again it is of interest to note that ARE(Q,, -n log V; 1; r) = ARE(V, T2; 1; r)2.
Using the facts that limrno r(a/v)/r(b/v) = b/a and limv v/F(b/v) = b, we also
have that
2
ARE(Q, -n log V;oo; r) = r


Notice that for r = 1 (the bivariate case),

ARE(Q,,-nlog V; 0.5; 1) = 1,

ARE(Q, -n logV;1; 1)=


and


ARE(Q, -nlog V;oo;1)= 1,


which all agree with the values in Table 1.1 and Table 1.2 (v = 0.5 corresponds to
the Laplace, v = 1 to the normal, and v = oo to the uniform distribution).
As indicated earlier, what follows are several graphs depicting the nature of the

Pitman ARE given in Theorem 3.2.4 for various values of v = vi = v2. Clearly, Q,







does very well when v = 0.1 (Figure 3.2). Of course, although we can compute and

graph the Pitman ARE at the values r1 = 1 and r2 = 1, the expression for the ARE

is not valid, so the graph should be ignored in those areas. Q, and -n log V perform

almost equivalently when v = 0.5 (Figure 3.3), and -n log V beats Q, when v = 1
and v = 10 (Figures 3.4 and 3.5).












1

0.75

0.5

0.25

0 ---


Figure 3.2. 1/(l+ARE(-n log V, Q,; v = 0.1))

















1

0.75

0.5 5



4 4 5
3 3
r2 2 2 rl

1







Figure 3.3. 1/(1+ARE(-n log V, 1,; v = 0.5))


Multivariate t-distribution family


The multivariate t-distribution indexed by its degrees of freedom df, is another
convenient elliptically symmetric class of distributions. Again we can evaluate the

expression for ARE(Q,,, -n log V) when the underlying distributions of X(') and X(2)























1

0.75

0.5
0.25


Figure 3.4. 1/(1+ARE((Q, -nlogV; v = 1))






















1

0.75

0.5

0.25
0


Figure 3.5. 1/(1+ARE(Qn, -nlog V; v = 10))







have varying tail weights. When df = 1, this corresponds to the multivariate Cauchy
distribution and as df -+ oo, the distribution approaches multivariate normality.
Let

9k (t) -(l) df /2


and

P (df f+ rk
Ck =
(7rdfk)rk/2r fk


Then X(k) has a multivariate t-distribution located at the origin with dispersion

parameter Irk and degrees of freedom dfk (X(k) ~ t(dfk)). The density function

of (R(k))2 is given by:

1 y l /2-I -(dk +'rk)/2
hk(t) = (3.6)
rTk dfk dfk dfk
dfk 2 ;


where B(-,-) represents the beta function. Thus (R(k))2 has the same distribution
as dfk(1 U)/U where U is distributed as beta with parameters rk/2 and dfk/2.
Before calculating the Pitman ARE, we derive a useful moment.

Lemma 3.2.5

b (rk + 2a dfk 2b -2a)

2' 2
EHO [((R~))2)(I_ (R(k))2bdf ] f B rk l2a- ,dfk- 2b--2a
B -T,2


provided rk + 2a > 0 and dfk > 2(a + b).






Proof of Lemma 3.2.5 Let T have the density h given in (3.6). Then


(1+


Tb]
d fk


1 y00
1 (t/dfk)T/(2-*+a(l + t/df)-(dfk+rk)2+bdt
dB( 'k dfk, \)
dfkB (2 2


1 jO trk/2-1+a(1 + t)-(df+Tk)/2+bdt
2rk dfk 2
B^T


1 o
= T t(,+2a)/2-1 +
B T rk dfk\ 0
B-2 T


B (rk + 2a dfk 2a 2b)
2 2
B k( d df,\
2' 2

so long as rk + 2a > 0 and dfk > 2(a + b). O

Theorem 3.2.5 If X(k) ~ t(dfk), with dfk > 4, and we assume that M1 = M', then


ARE(Q,, -n log


1
V;df, df2)
r1 r2


r, r2
(r( + 1 (r2 +
F p)~ z~)


+ r (df-) 1)r


f( \2


df2 +1
2 )


dflm 1/2 2
df2


Proof of Theorem 3.2.5 Since


dfk + rk
2dfk


-1


,2(d)


.____


E WJ
f(WT


t)-((rk+2a)/2+(df;-2a-2b)/2)dt


Ok(t) = 9'. (t)lgk (t) =







E, = Eo [R')] E,, [IR'),((R('))2)]

f EHo IR(") EHo[ (o
+2d f, I\ f +1 df+

if1 B (?-2 +Idf2- 1) + f ( df 1)I
+f 7+rd2 2 2j 2 2
^+n d ^d
2df= df
2dmB B (")
2' 2 2 2


















the marginal fourth moments must exist for -n log V to have a limiting distribution,
we need df > 4. Putting these 2 for and 2 in (3.3) gives the result.



















Corollary 3.2.2 If df1 =df.2 df andR r = r2 = r then
2 fARE( df,2 dfd
B f ) B
22 2 2 2


ri t pri ftred it t it A uder f-1m
tdfivare 2 2t a q ol 2 2d,


and similarly,



r -r -r (d-)r ( 12
2 2 2 2


The moment conditions of Lemma 3.2.2 are easily seen to be satisfied if dfk > 2, but as
the marginal fourth moments must exist for -n log V to have a limiting distribution,
we need dfk. > 4. Putting these expressions for pj and 02 in (3.3) gives the result. D

Corollary 3.2.2 If df, = df2 =- df and ?ri = r2 = r then


ARE(Q,, -n log V;df; r) =-2)r(22- (3.7)
2 2

Interestingly, the expression (3.7) can be factored into the Pitman ARE under mul-
tivariate normality times a quantity involving only df, which (obviously) goes to one







as df -- oo. Thus, for example, putting df = 5 yields


ARE(Q, -nlog V;5;r) = 8 22 7
r\


where 1024/8172 = 1.2809 or putting r = 1 yields
Sdf + 1 df 1
42 2
ARE(Q,,, -n log V; df; 1) = 2 2
7r2 (df


Again we include some graphs of the Pitman ARE given in Theorem 3.2.5 for
various values of df df1 = df2. We could calculate the Pitman ARE when df < 5,
but since -n log V is guaranteed to have a limiting distribution only when df > 5,
the value would be meaningless. Hence we consider only df > 5. Of course, we

anticipate that Q, is vastly superior to --n log V when df < 5, but we do not have a
way to quantify their relative performance in this instance. (We do include df = 1
in the simulation study, which will demonstrate if our intuitive feeling is borne out.)

When df = 5 (Figure 3.6), we see that Q, and -n log V perform essentially the same.

When df = 10 and df = 100 (Figures 3.7 and 3.8), -nlog V beats Qn. In fact, for
df = 10, the Pitman ARE is already only 1.057 times that of the multivariate normal
Pitman ARE, for which -n log V is optimal.






















1

0.75

0.5
0.25


Figure 3.6. 1/(l+ARE(-n log V, Q,; df = 5))






















1

0.75


0.25


Figure 3.7. 1/(1+ARE(Q,,, -nlog V; df = 10))






















1

0.75

0.5
0.25

01


Figure 3.8. 1/(1+ARE(Q,, -nlog V; df = 100))







3.3 Model 2


A model proposed by Puri and Sen (1971) is given by

( (') (Y) + AZ(l)
X(2) Y(2) + AZ(2)

y ( l) + A Z(Z ')
y(2) ) Z(2)

= Y + A Z


where Y(), y(2) and Z are mutually independent, and the matrix CovH. [Z() Z(2)]

consists entirely of nonzero elements (although it appears that it is sufficient for it not
to be the zero matrix). They state that such a model may prove useful in analyzing
group tests in psychology. For example, the outcomes of two reading tests and two
math tests can be described by a (linear) combination of individual group factors
pertaining to the reading or mathematical abilities and common factors corresponding
to intelligence or comprehension. In general, the distribution of X is determined by
using a convolution formula since it is a sum of two independent random vectors.
However, obtaining a closed form expression for the density function, an integral
step in being able to work out the details related to contiguity, is typically non-
trivial. An exception is when Y(), y(2) and Z have multivariate normal distributions.

Thus, if Y(1) ~ MVN(0, I,), Y(2) ~ MVN(0, I,,), and Z ~ MVN(O, ), then
X MVN(O, I,+2, + A2E).
Since the determination of contiguity and computation of the limiting distribu-
tions of -n log V, Q,,, and -n log S'J under this sequence of multivariate normal
alternatives is virtually identical to the computation under alternatives described by







Model 1, the details will be omitted. The end result is that

-n log V X X, (Agvec (S12)' vec (12)) ,

jd -2 4( )
-n log SO X (Avec (E2)' vec (E2)


and



n X 12r 2 (f 2e A vec (E12)' Vec (12)
ri r21 7 rr, (r2


From these expressions, we observe that the Pitman ARE's will be identical to those
derived from Model 1 when the underlying distribution is multivariate normal. This
is further indication that the actual form of the model used is somewhat irrelevant
when dealing with local alternatives.












CHAPTER 4
MONTE CARLO STUDY


4.1 Methods


All simulation programs were written in the C programming language. Previously

written routines acquired from various sources were combined with original source

code to complete the main procedure. Outside sources included a large archive of soft-

ware maintained by AT&T called Netlib and another

archive of statistically related software maintained by Michael Meyer at Carnegie

Mellon University called Statlib . Included in the non-

original code used are parts of the following libraries: c/meschach-a set of functions

which do numerical linear algebra, dense and sparse, with permutations, error han-

dling and input/output by David E. Stewart , c/cephes--

a set of special math functions and IEEE floating point arithmetic by Stephen L.

Moshier and ranlib-c-a set of random variate gen-

erators translated from FORTRAN by Barry Brown .

Also used are chisq. c, f. c and z. c, which are functions written by Gary Perlman to

compute probabilities and percentiles of the chi-square, F and normal distributions

and Ll.f, a FORTRAN routine based on an algorithm by Barrodale and Roberts

(1974) to compute the least absolute value solution to an over-determined system of

equations (personally translated to C). The final program was compiled using gcc

(GNU project C compiler v2.4) on a SPARC 10. Several programs which are of

interest are included in Appendix C.







The two distribution types used were the exponential class and multivariate t de-

scribed earlier. The method for generating observations from these distributions has

three parts (see Johnson, 1987). First, a vector of iid N(0, 1) random variables is

generated. Second, the vector is divided by its Euclidean norm, which results in a

vector uniformly distributed on the unit hypersphere. Third, multiplication by a pos-

itive random scalar with the appropriate distribution (va-kGamma(r/(2Vk), 1)1/(2"k)

for the exponential class and V/dfk/x for the multivariate t) yields the desired

multivariate observation. Model 1 (3.1) was then used to generate the dependence

structure. For ease of comparison, we restricted the study to cases where 7r = r2 r

and the underlying distribution types were identical for each set of variables. Specif-

ically, for the dimensions r = 1,2, we used the distributions v = 0.1,0.5, 1, 10 in the

exponential class and df = 1,5 in the multivariate t. For r = 3, we considered only

the exponential class with v = 0.1,0.5, 1. The sample size, n, was kept at 30 and

the number of repetitions at each setting was 2500 when r = 1,2. Because of the in-

creased computing time needed when r = 3, the number of repetitions was decreased

to 1000. To gauge the precision of the empirical powers computed, we provide a table

summarizing the maximum estimated standard errors of the empirical power over

various levels of the true power.

4.2 Statistics Compared


Of course -n log V, -n log SJO, and Q, were included in the study, but numerous

other normal theory tests not explicitly investigated in this thesis were also added to

judge ,, against. In particular, the standard multivariate tests used in the SAStm

procedure PROC CANCORR and their variants were considered. These tests are

all based on the sample canonical correlations (/i,..., r)), where A > > 2,








Table 4.1. Maximum Estimated Standard Errors for Empirical Power
Repetitions
Power 1000 2500
0.00 0.05 0.0069 0 0044
0.05 -0.10 0.0095 0.0060
0.10- 0.25 0.0137 0.0087
0.25 0.75 0.0158 0.0100
0.75 0.90 0.0137 0.0087
0.90 0.95 0.0095 0.0060
0.95 1.00 0.0069 0.0044



are the eigenvalues of S Al1A12A2-A12. Several of the tests can be approxi-

mated both by an F distribution and a chi-square distribution. We present both

forms for comparison. First is -nlog V, which multiplied by the Bartlett correc-

tion factor 1 (r, + 7'2 + 3)/(2n) (Box, 1949) is labeled as LX. The F approxima-

tion (a different transformation of V) is naturally labeled LF. VF and VX are F

and chi-square approximations, respectively, for Pillai's trace CI ii2. Likewise UF

and UX are F and chi-square approximations, respectively, for the Lawley-Hotelling

trace = 1, ?/(1 k). The last normal theory test is RF, an upper bound on an F ap-

proximation to Roy's greatest root //(1 2). For consistency, we label -n log S'J

as PS.

The Oja median (Oja, 1983) was used in Q,, which we will henceforth call QI, to

estimate the nuisance parameters 08 and 02. This generalized median, which is the

point minimizing the sum of the volumes of all p-dimensional simplexes formed from

p 1 sample points and itself, is equivariant under the group g and asymptotically

normal with rate n (Oja & Niinimaa, 1985). When p = 1, the Oja median is just







the usual univariate median. The method used to compute it is based on the Li-norm

formulation of the minimization as given in Niinimaa (1992).

The final statistic, Q2, is defined as

Q2 1, ()'2 (1) (2)' (2)
Q2 = i J U i ,I
Si=1 j=l

where
11 /2(x(O
_o) 0x (Xi-e0)
-1)
(X() 0 )' x (X') 0b)


and

(2) 22 (x) 2)
(X 02)'22 (X(2 02)

and was included because it is asymptotically equivalent to Q1, but is much simpler

computationally. Robust M-estimates of ESa and E22 as described by Randles, Brof-

fitt, Ramberg, and Hogg (1978) were used in Q2 as well as the same Oja median

statistics used in Q1 to estimate 01 and 02. Thus Q2 is invariant under g and it is

hoped that it will be as robust as Q1.

4.3 Results


The outcome of the Monte Carlo study is presented graphically in a series of figures

at the end of the chapter. However, the values used in these figures are in Appendix C

in tabular form as well. In order to facilitate understanding of the simulation results,

we make some general comments for specific cases.

When r = 1 (the bivariate case) the results are not unexpected. Since Q1 and Q2

are essentially equivalent to q' (a very robust statistic), it is easy to understand

why, for the heavy-tailed distributions, (see Figures 4.1 and 4.5 for graphs of v =

0.1 and df = 1) both Q1 and Q2 do better than the normal theory tests. It is







important to keep in mind that by saying "better", we mean that, not only does the

test have higher power over the alternative, but that the test has at least come close

to maintaining the designated nominal level of 0.05. For tests that do not achieve the

latter criterion, it is difficult to compare them with competing procedures. In general

however, it seems prudent to be biased in favor of procedures which maintain the

desired level versus procedures which have higher power but do not maintain the the

nominal level. Thus although there are instances where Q1 and Q2 are "beat" by other

procedures at various points in the alternative, for the most part Q1 and Q2 are much

better at maintaining the designated 0.05 level, indicating their favorability. This is

the case here, and in general, for heavy-tailed distributions like the multivariate t with

df = 1. An interesting observation is that although PS is asymptotically equivalent

to q', it doesn't do as well as Q1 and Q2 for these distributions. In fact, it is uniformly

worse (conservative) than both Q1 and Q2. One reason may be that the central limit

theorem doesn't work quite as fast on the log of a sum (like PS) as it does on a strict

sum of terms (like Q1 and Q2). We note that in this case the normal theory tests

perform almost identically, with the exception of UX, which is too liberal. Hence for

r = 1 we do not differentiate among the normal tests, except to exclude UX. As the

distributions become lighter tailed (see Figures 4.2, 4.3, 4.4, and 4.6 for graphs of

v = 0.5, 1, 10 and df = 5), the normal tests are clearly better than than Q1, Q2,

and PS.

When r = 2, we examine the competitors showings in greater detail. Since RF

and UX are consistently above the 0.05 level by a large margin, while the rest of the

normal theory tests, although they may also exceed it, are less liberal and perform

comparably, we will not differentiate among the normal theory tests except to exclude

RF and UX. For v = 0.1 and df = 1, (see Figures 4.7 and 4.11), both Q1 and Q2

perform well, maintaining the 0.05 significance level and showing a steep increase

in power. PS comes close to doing as well, but overshoots the 0.05 level and has







power slightly below Q1 and Q2. None of the normal theory tests do very well. In

fact, for v = 0.1, they all have uniformly much lower power than Q1 and Q2 while

greatly exceeding the 0.05 nominal level. For v = 0.5 and df = 5 (see Figures 4.8

and 4.12), QI and Q2 seem to do equally as well as the normal theory tests, with

PS again exceeding the 0.05 level and having power somewhat less than the others.

For v = 1, 10 (see Figures 4.9 and 4.10), Q1 and Q2 start off very competitive, but

have decreased power relative to the normal tests as the dependency is increased. PS

again is uniformly worse than the others, having exceeded the 0.05 level and having

lower power over the alternative.

When r = 3, the situation remains essentially the same. Referring to Figures 4.13,

4.14, and 4.15, RF and UX are uniformly bad with Q1 and Q2 doing slightly worse

than the other normal tests for v = 1 but competitive for v = 0.5 and dominant for

v =0.1.

In general, it appears that Q1 and Q2 do a much better job of maintaining their

nominal level than PS or any of the normal theory tests. Q1 and Q2 are consistently

very close to maintaining the 0.05 nominal level where the others vary widely either

above or below. This implies that the small sample applicability of Q1 or Q2 is very

good. We might add that most of the simulation results presented here are basically

in agreement with the Pitman ARE's derived in Chapter 3. For example, compare the

essentially equivalent performances of Q1 and LX when v = 0.5 with the associated

Pitman ARE (see Figure 3.3).





59








1.0- -

V F .-. .
0.9- UF
RF /
----- LX
0.8- VX //
UX /
............. PS
//

0.7 Q /
Q2 /



1/!
0.6 ,/

oi /I
S0.5- /

E i /
0.4 /"
0.4- //
//
//
0.3-


0. /
0.2- /




Nominal value

0.0-
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.1. r = 1, n = 30, v = 0.1, reps = 2500





60









1.0-
----- LF /
VF VF
0.9- ---- UF /~
-- RF
----- LX r'
0.8- --- VX /
--- UX //
PS i/
............. PS

0.7- Q1 /
Q2 ./
//
0.6-
0.6 // //
// "

0.5 /-
g0o // .."

E
0.4- // /
/.









Nominal value
0.3 -
/ .
/ /

0.2 //
// .


Nominal value

0.0
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.2. 7 = 1, n = 30, i = 0.5, reps = 2500
















1.0-



0.9-



0.8-



0.7-



0.6-



0.5-



0.4-



0.3-



0.2-



0.1 -



0.0 -


Nominal value


I
0.05


I
0.10


0.15

Delta


I
0.20


I
0.25


Figure 4.3. r = 1, n = 30, v = 1, reps = 2500


----- LF
----- VF
UF
RF
----- LX
----- VX
UX
............ PS
Q1



Ia/
//




1/
//
/t
/ /'
1/
//
/I,/
/!


// / ...


/ / ../

.o'7'


I
0.30







62









1.0- -

----- VF
0.9- ---- UF
RF A/
---- LX '
0.8 ---- VX
SUX /
... PS .. //
0.7 Q1 /
Q2 //
//
0.6 /


S0.5 /
lt
w//
//












0.0 -
0.3 /.5 .







tDelta
// ;/

0.2 ,.. .
//
0.1 -,, .
--s- ='-'^-'____ Nominal value

0.0


0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.4. r = 1, n = 30, v = 10, reps = 2500




63









1.0-
L------ LF. ..--
----- VF /
0.9- --- UF / .
-- RF R
----- LX
0.8- VX /


0.7- Q1 /
//
Q2 //

0.6- / /
S//
S0.5-
" 0-"- //
//
E
w/


///
0.4-


0.3 //





0.1 /// .-
0.2


0.1 /
Nominal value

0.0-
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.5. r = 1, n = 30, df = 1, reps = 2500




64









1.0-
-----LF /
---- VF
0.9- UF
RF
LX
0.8- ---- VX /
UX /i
............. PS / "
PS //
0.7 Q1 /
Q2 //
//

0.6- /


0 //
S/ /

0.4 -




S0.- / -0
/ /







Nominal value
1/
0.2 ///








0.0-




0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.6. r = 1, n = 30, df = 5, reps = 2500




65









1.0-
----- LF
----- VF
0.9 ----- UF
RF
----- LX

0.8- --- VX
UX -//U
............. PS
0.7- Q1
Q2 //


0.6 -




EI. //
0.5-

E
0.4- //


0.3-/


0.2-


0.1 -
Nominal value

0.0-
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.7. r = 2, n = 30, v = 0.1, reps = 2500















1.0
----- LF---
----- VF
0.9- UF
RF
----- LX '

0.8 VX V
SUX /
............. PS

0.7- 1
Q2


0.6- //

o ////
0.5 -



0.4



///






0.1-
0.2- / / /




Nominal value

0.0

I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.8. r = 2, n = 30, v = 0.5, reps = 2500















1.0-



0.9 -



0.8 -



0.7-



0.6 -



0.5-



0.4-



0.3 -



0.2 -



0.1 -



0.0 -


I
0.15

Delta


I
0.20


I
0.25


I
0.30


Figure 4.9. r = 2, n = 30, v = 1, reps = 2500


Nominal value


I
0.05


I
0.10


1 I




68









1.0-
----- LLF-
----- VF
0.9- ---- UF
RF
----- LX
0.8- ---- VX
UX--- /
............ PS// / /
0.7- Q1 ///
0.7 -

S2//
Q2 // / /
0.6-

o /, / /
a,
c60.5- /









S2 ','/
0.4-



0.3- /

//
0.2 -


0.1
0 .1 -------------------....

Nominal value

0.0-
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.10. r = 2, n = 30, v = 10, reps = 2500




69








1.0 _
LF
----- VF
0.9- UF /
-RF


0.-- u// 7/
............. PS /
0.7 Q//

Q2//


a/ 5
7 0.5- // /

E
0.4


0.3- /


0.2-


0.1-
Nominal value

0.0-
I I I I I I I
0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.11. r = 2, n = 30, df = 1, reps = 2500




70









1.0-
----- LF
----- VF
0.9- --- UF
RF
----- LX
0.8- --- VX
UX /
............. PS

0.7- Q1
Q2 /


0.6 -

I/ :/



0.4/



0.3- /





I I /I







0.0 0.05 0.10 0.15 0.20 0.25 0.30

Delta


Figure 4.12. r = 2, n = 30, df = 5, reps = 2500

















----- LF
----- VF
----- UF
--- RF
----- LX
----- VX
SUX
Q1
Q2


1.0-



0.9-



0.8-



0.7-



0.6-



0.5-



0.4 -



0.3-



0.2 -



0.1 -



0.0 -


I
0.05


I
0.10


I
0.15

Delta


I
0.20


I
0.25


Figure 4.13. r = 3, n = 30, v = 0.1, reps = 1000


Nominal value


I
0.30


















----- LF
----- VF
----- UF
RF
----- LX
----- VX
--- UX
-Q1
Q2


1.0-



0.9 -



0.8 -



0.7 -



0.6 -



0.5-



0.4 -



0.3 -



0.2 -



0.1 -



0.0 -


/
/
/
/


I,
II
'I
I,
rI I


Nominal value


I
0.05


0.10
0.10


I
0.15

Delta


I
0.20


I
0.25


Figure 4.14. r = 3, n = 30, v = 0.5, reps = 1000


/
/
/
/


I
0.30



















1.0-



0.9-



0.8-



0.7-



0.6-



0.5-



0.4-



0.3-



0.2 -



0.1 -



0.0 -


I

I.
I


I
0.05


I
0.10


Nominal value


Figure 4.15. r = 3, n = 30, v = 1, reps = 1000


----- LF
----- VF
----- UF
--- RF
----- LX
----- VX
SUX
Q1
-Q2




/
/
/
/


/ //


I
0.15

Delta


I
0.20


0.25
0.25


1
0.30












CHAPTER 5
APPLICATIONS


5.1 Analysis of Newborn Blood Gas Data


The State of Florida mandates that measurements be taken of newborn blood at

birth. This is usually done from the umbilical cord, but it is unclear whether practi-

tioners routinely get umbilical arterial blood or umbilical venous blood. It is further

unclear whether these values are related to a simple arterial draw from the abdomen

that can be done one hour later in the nursery. Behnke, Eyler, Conlon, Woods, and

Thomas (1993) investigate the use of birth weight, gestational age, Apgar scores, cord

blood gas values, and first arterial blood gas values as diagnostic criteria for perinatal

asphyxia and subsequent low neurodevelopmental outcome in very low birth weight

infants. Their data consist of 57 cases for which good blood gas information exists

at all three sites-umbilical venous at birth (UV), umbilical arterial at birth (UA),

and abdominal arterial one hour after birth (AA). For each site we consider three of

the blood gas measurements taken: bicarbonate (HCO3), partial pressure of carbon

dioxide (C02), and partial pressure of oxygen (02).

We are interested in determining if there is any relationship between the UA and

UV blood draws at birth and also if either of these are related to the AA blood

draw. Intuitively, since the UA blood originates from the mother and the UV blood

from the newborn, we expect that there should not be as much dependence between

UV and UA, and UV and AA blood, as there is between UA and AA blood. If

these conjectures are supported by the data, there is need for concern with respect to

the methods of the practitioners. Practically speaking, the goal of mandatory blood







draws on newborn babies when the diagnostic potential of the measurements may

well depend on the technique used seems a flawed idea at best.

A summary of the tests of independence is in Table 5.2. Notice that all the normal

theory tests detect dependence in each of the three comparisons, while the nonpara-

metric tests only indicate dependence between the UA and AA measurements. In

fact, there is strong agreement among all tests that there is dependence between the

UA and AA blood gas measurements. A visual inspection of the blood gas measure-

ments however (see Figures 5.1, 5.2, and 5.3 for plots created by xgobi-a program

which allows three-dimensional rotation of the data), reveals three observations in

the UA determinations which are quite distant from the remainder of the data.

After deleting these observations (28, 52, and 55), the tests were re-run. The

results are fairly telling. The normal normal theory tests involving the UV blood

gas measurements are now non-significant, with the exception of RF. Of particular

interest is how drastically the p-values of these tests changed with the deletion of

just three points out of 57. This would seem to be an undesirable property of any

test. Although PS managed the same conclusions as Q1 and Q2 (with and without

the outliers), its p-value also changed significantly making it suspect also. Only Q1

and Q2 remained largely unaffected by the removal of the outliers.

Although this data is useful in illustrating Q,'s resistance to outliers, it is deficient

in that there is structure present which is ignored by all these statistics. HC03, CO2,

and 02 are repeated in each set of variates, but because of the invariance of the

statistics with respect to the labeling of the variables within a set they fail to take

advantage of this fact. Thus, there are probably other techniques for analyzing this

data which might be more powerful. The next section illustrates an example where

there is no such natural pairing.







Table 5.1: Newborn Blood Data


Obs UV UA AA
HC03 CO2 02 HCO3 CO2 02 HCO3 CO2 02


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41


22.8
19.8
24.2
16.9
13.0
14.1
24.2
17.6
21.8
18.4
16.4
19.8
22.8
17.2
13.2
23.9
20.6
22.5
15.1
16.8
24.8
23.0
19.5
20.9
19.6
22.1
17.0
16.8
19.6
17.0
18.2
22.4
27.0
21.5
19.7
23.1
22.2
18.8
11.4
21.1
23.6


37
28
45
18
19
26
48
39
68
99
23
21
42
16
21
94
34
29
25
21
63
26
30
39
38
53
28
29
34
26
18
20
49
50
58
41
47
21
9
36
47


246
80
44
171
83
70
40
81
34
67
110
63
73
365
280
66
169
207
66
434
45
49
74
39
48
48
85
47
194
283
64
271
36
56
35
225
60
173
87
131
198


22.4
16.4
22.0
23.3
22.3
22.4
19.3
21.8
15.6
22.1
23.5
22.1
23.2
21.4
16.9
19.7
25.7
13.5
17.0
25.4
26.0
21.8
24.8
22.2
24.9
24.5
27.7
21.0
24.5
19.5
16.6
20.5
26.6
22.8
23.0
24.7
22.3
21.6
21.2
19.5
21.9


40
39
55
48
46
46
39
34
51
43
50
47
48
42
43
42
54
27
62
47
58
40
50
42
45
42
70
111
57
44
56
38
50
51
46
45
58
51
46
37
45


9
21
9
21
29
26
33
16
19
18
39
31
17
13
12
30
23
20
14
21
21
14
17
20
7
3
11
20
21
22
24
10
20
22
19
14
9
30
10


23.9
18.4
27.0
21.8
21.2
21.5
22.0
21.1
19.3
20.3
22.8
23.2
23.0
22.8
19.0
19.0
22.3
22.5
20.6
22.9
25.0
22.9
24.4
20.0
23.3
23.2
24.8
20.3
26.4
18.6
21.2
21.3
25.5
20.6
22.4
22.2
20.7
25.3
26.2
19.3
21.8


40
36
53
39
37
40
42
29
44
28
40
38
44
41
35
34
43
42
39
39
49
38
45
35
38
36
44
103
50
33
47
35
45
42
40
37
45
49
54
35
42


36
43
15
31
23
30
30
41
32
36
25
27
52
37
27
26
23
39
64
51
23
29
25
20
29
31
30
6
27
28
31
32
34
24
38
27
27
25
14
28
15







Table 5.1: -continued


5.2 Analysis of Fitness Club Data


The section of SAS/STAT User's Guide, Volume 1, which describes the SAS pro-

cedure PROC CANCORR has an example that uses data provided by Dr. A. C. Lin-

nerud, North Carolina State University, in which three physiological variables and

three exercise variables were measured on twenty middle-aged men in a fitness club.

It demonstrates how PROC CANCORR can be used to determine if the physiological

variables are related in any way to the exercise variables.

Here again we have potential outliers, which we can see from examining Figures

5.4 and 5.5. A summary of the statistical analysis of this data (with and without

observations 10 and 14) is in Table 5.4. Interestingly, the p-values for the normal

tests all decrease while the p-values for the nonparametric tests all increase with the

removal of the outliers.


Obs UV UA AA
HCOs CO2 02 HC03 CO2 02 HCOs CO2 02
42 18.8 44 46 20.3 57 13 21.0 53 19
43 12.3 12 300 18.7 45 15 20.2 41 15
44 13.9 12 270 18.7 45 8 22.8 50 22
45 21.9 42 194 19.7 58 23 19.6 53 26
46 16.6 27 27 20.4 42 30 19.6 38 70
47 19.4 27 70 22.7 42 16 22.4 34 23
48 21.3 28 38 23.9 49 15 23.9 46 21
49 20.8 44 299 21.9 43 24 18.7 32 40
50 21.4 42 163 24.3 51 16 23.3 44 25
51 19.6 38 199 21.6 41 29 21.3 33 40
52 26.2 37 135 25.6 23 51 25.6 44 56
53 16.1 26 40 21.1 35 25 20.2 30 35
54 19.5 37 194 22.2 56 14 19.4 42 26
55 11.1 27 42 16.9 115 11 17.2 96 18
56 15.2 25 94 19.6 46 19 18.6 32 28
57 22.2 50 76 21.3 38 25 22.0 38 28








Table 5.2. Statistical Analysis of Newborn Blood Data
UV vs. UA UV vs. AA UA vs. AA
Statistic Value P-value Value P-value Value P-value
LF 2.5649 0.0097 2.1290 0.0317 35.6598 0.0000
VF 2.3596 0.0158 2.0662 0.0356 22.0281 0.0000
UF 2.7189 0.0058 2.1519 0.0285 45.9539 0.0000
RF 7.8735 0.0002 5.2465 0.0030 122.2455 0.0000
PS 12.8302 0.1704 7.0634 0.6305 70.6444 0.0000
Q1 4.9852 0.s)56 9.6629 0.3785 82.9273 0.0000
Q2 5.2102 0.8156 8.1707 0.5170 82.5464 0.0000
after the outliers are removed
LF 1.1296 0.3476 1.0216 0.4271 14.7553 0.0000
VF 1.1137 0.3564 0.9938 0.4477 12.8989 0.0000
UF 1.1386 0.3397 1.0446 0.4080 14.5733 0.0000
RF 3.2089 0.0308 3.2942 0.0279 29.3290 0.0000
PS 6.4742 0.6917 4.9757 0.8364 59.2939 0.0000
Q1 3.2630 0.9530 9.4586 0.3961 75.7127 0.0000
Q2 3.8196 0.9229 7.4735 0.5880 75.4526 0.0000


5.3 Analysis of Cotton Dust Data


Merchant et al. (1975) studied the effects of cotton dust exposure on human

beings by measuring several respiratory variables and several blood-related variables

on 12 subjects exposed for six hours. The data consist of changes in these variables

from baseline. It may be of medical interest to determine if these two sets of variables

are independent or not. Included among the respiratory variables are closing capacity

(CC), vital capacity (VC), and total lung capacity (TLC). Two blood-related variables

are oxygen (02) and white blood count (WBC).









Table 5.3. Fitness Club Data
Obs. Weight Waist Pulse Chinups Situps Jumps


Table 5.4. Statistical Analysis of Fitness Club Data
Outliers In Outliers Out
Statistic Value P-value Value P-value

LF 2.0482 0.0638 2.4847 0.0305
VF 1.5587 0.1551 1.5877 0.1504
UF 2.4938 0.0238 3.4540 0.0045
RF 9.1986 0.0009 13.5137 0.0002
PS NA NA 15.4118 0.0802
Q1 12.9838 0.1633 11.8387 0.2226
Q2 13.4917 0.1416 12.5776 0.1827


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20


191
189
193
162
189
182
211
167
176
154
169
166
154
247
193
202
176
157
156
138


162
110
101
105
155
101
101
125
200
251
120
210
215
50
70
210
60
230
225
110


60
60
101
37
58
42
38
40
40
250
38
115
105
50
31
120
25
80
73
43






























UV_HC03



UV_02



52





. > *


UV_C02


*
*










528


Figure 5.1. Umbilical Venous Blood Gas Measurements





81






















UA_02


UA_HCO3












S. e52


*. r.

,55 .0 .

0
o *


Figure 5.2. Umbilical Arterial Blood Gas Measurements





























AA_02


AA_HC03





AA_C02





,52
*





55 *
... .


Figure 5.3. Abdominal Arterial Blood Gas Measurements


~





83





















Waist





,10 .

S* Pulse 14





Weight
Weight


Figure 5.4. Physiological Measurements






















Jumps


Situps


Chins


Figure 5.5. Exercise Measurements











Table 5.5. Cotton Dust Data
Obs. CC VC TLC 02 WBC
1 -4.3 0.24 -0.11 -1.0 6000
2 4.4 -0.29 -0.01 -1.5 -350
3 7.5 0.10 0.67 1.0 -250
4 -0.3 0.13 0.31 6.5 1675
5 -5.8 0.02 -0.75 -3.0 875
6 14.5 0.48 1.14 -3.0 -100
7 -1.9 -0.05 -0.22 -15.5 1075
8 17.3 -0.62 0.62 -13.5 1675
9 2.5 -0.16 0.12 0.0 1500
10 -5.6 0.15 -0.14 -4.0 2200
11 2.2 0.25 0.40 -2.5 650
12 5.5 -0.42 0.22 -1.0 3025


Table 5.6. Statistical Analysis of Cotton Dust Data
Statistic Value P-value
LF 0.8623 0.5452
VF 0.9339 0.4976
UF 0.7783 0.6024
RF 1.6581 0.2521
PS 2.4917 0.8694
Q1 14.7390 0.0224
Q2 13.4729 0.0361












CHAPTER 6
CONCLUSION


6.1 Discussion


With the increasing availability of very fast computers, multivariate and nonpara-

metric procedures which would have been impossible for the average statistician to

implement several years ago can easily be utilized today. This fact is responsible

for the explosion of such multivariate and nonparametric methodologies seen cur-

rently. The goal of this dissertation has been to research one such methodology, the

interdirection quadrant statistic (Qn), in testing for independence between two sets

of variates. The basic competitors considered were Wilks' likelihood ratio criterion

(-n log V) and a specific member of a class of statistics invented by Puri and Sen

(-n log S'J). As demonstrated in Chapter 3, the Pitman ARE's indicate that Q^
does quite well relative to -n log V for heavy-tailed distributions and is competitive

for moderate-tailed distributions. The statistic Q,, under spherical alternatives, ap-

pears to be uniformly better than its natural nonparametric competitor -n log SO.

Simulation results concur with theoretical findings in the sense that the empirical

powers of the competitors are ordered in the same way as the Pitman ARE's indicate

they should be.

6.2 Further Research


Other avenues of research might be to investigate the potential of the interdirec-

tion quadrant statistic (i) for describing the nature of the association between two sets







of variates instead of merely using it as a test of independence, or (ii) for determining

which variables in a set may or may not be contributing to an association between

the two sets. It might also be desirable to make comparisons with other members of

Puri and Sen's class of statistics, like a multivariate analog of Spearman's rho. Com-

puting issues that arose during the simulation study also lead to possible research

areas. Since the time to compute the interdirections (based on simple looping algo-

rithms) is on the order of nk, where k is the dimension, for even moderate sample

sizes, use of the interdirections becomes impractical. Possible work-arounds to this

limitation might be (i) to find a suitable approximating statistic (which might entail

more simulation work), (ii) to derive a faster algorithm for computing the interdirec-

tions, or possibly (iii) to estimate, by using some sampling method for instance, the

interdirections. Of course, since the interdirections have wider applicability than the

present independence testing situation, these results would naturally have broader

appeal than this context. The statistic Q2 defined in Chapter 4 and included in the

simulations there, is an example of a preliminary step in examining the feasibility

of (i). It appears to do quite well, so that this seems to be a promising idea.