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
Chisquare 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 chisquare distribution under the null
hypothesis of independence when each set of variates is elliptically symmetric, mak
ing it asymptotically distributionfree. The new statistic is compared to the classical
normal theory competitor, Wilks' likelihood ratio criterion, and a componentwise
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 heavytailed dis
tributions and is uniformly better than the componentwise 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 heavytailed 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 wellknown 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 Pitmantype 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 = n1/4Ao. 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 realvalued random variables are independent, we
consider testing whether two vectorvalued 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 A2AA' 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'A12AAA'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 AA2A2 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 Componentwise 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
componentwise 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 componentwise ranking scheme is that they do not have the desirable invariance
property exhibited by V. Thus ,S', using componentwise 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 realworld 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(n1/2). 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 distributionfree 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 nI1vec(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(n1/2), 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 realvalued
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 twofold. 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 (IAA1) 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 positivedefinite
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 = n1/2Ao, where
Ao > 0, is contiguous to the null hypothesis. To achieve this we follow the rationale
of Hajek and Sidak (1967, pp. 201214), 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 firstorder
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. 210213) and a univariate scale alternative (pp.
213214) 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 (JAA1) + abs (AA s ') a fY (A'z
A \ fy (x) WA fy (X)
Now
abs (IAAi') = abs (JAA1) tr (AtP),
where
9 I,, Mi)
P = AA =
A \M2 I2
abs (AA )
A=0
= abs (IAol1) 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. = nV 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
n1/2 ^ 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 Ustatistics, 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(n1)
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(n1)
= 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)
= nvec 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 Tlieranm 3.2.2 Let
S, = n1'/vec (B)' vec X} ')X 2)
Sn1/2 X')BX!2)
= n'/2 U!1)'(R 1) 2B) 2)
i=1
The key quantity is again
12 = 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 = n1 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
= n1vec sgn (U 1) sgn (U )vec sgn (U') sgn (U2)' + o (l)
= 71P\i=lkl
= n1 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 (affineinvariant) interdirection sign test with a (nonaffineinvariant) component
wise sign test, spherical symmetry is favorable to the componentwise test. He further
states that although it might be possible to improve slightly the efficiency of the
componentwise 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 rkdimensional 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 77k2 COS 77r1
((k) = sin i sin 772 ... sin 7k3 cos 77r2
k) = sin 791 sin 772 .. sin r/74 COS 7,r3
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'k2 771 sin5k3 772 *.. sin 77r2,
where 0 < j7i < nr, i = 1,...,rk 2 and 0 < 7rk1 < 27r, then U(k) will have the
uniform distribution on the unit hypersphere of dimension rk. Clearly the angles are
independent, with 77rkl 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.
k1
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 zrk2dz
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, = n1'2 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 21 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+)/2l 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/i1)/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 > (rk2)/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+2v1 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 tdistribution family
The multivariate tdistribution 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 tdistribution 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 /2I (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 2b2a
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/21+a(1 + t)(df+Tk)/2+bdt
2rk dfk 2
B^T
1 o
= T t(,+2a)/21 +
B T rk dfk\ 0
B2 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;2a2b)/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 f1m
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/meschacha 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 ranlibca 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 chisquare, 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 overdetermined 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 (vakGamma(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 Al1A12A2A12. Several of the tests can be approxi
mated both by an F distribution and a chisquare 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 chisquare approximations, respectively, for Pillai's trace CI ii2. Likewise UF
and UX are F and chisquare approximations, respectively, for the LawleyHotelling
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 pdimensional 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 Linorm
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 (Xie0)
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 Mestimates 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 heavytailed 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 heavytailed 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 sitesumbilical 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 xgobia program
which allows threedimensional 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 rerun. The
results are fairly telling. The normal normal theory tests involving the UV blood
gas measurements are now nonsignificant, with the exception of RF. Of particular
interest is how drastically the pvalues 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 pvalue 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 middleaged 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 pvalues for the normal
tests all decrease while the pvalues 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 Pvalue Value Pvalue Value Pvalue
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 bloodrelated 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 bloodrelated 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 Pvalue Value Pvalue
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 Pvalue
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 heavytailed distributions and is competitive
for moderatetailed 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 workarounds 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.
