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Interdirection tests for repeated measures and one-sample multivariate location problems

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Interdirection tests for repeated measures and one-sample multivariate location problems
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Jan, Show-Li, 1961-
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Population parameters ( jstor )
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Thesis (Ph.D.)--University of Florida, 1991.
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Includes bibliographical references (leaves 107-110)
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Typescript.
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Vita.
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by Show-Li Jan.

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INTERDIRECTION TESTS FOR REPEATED MEASURES AND ONE-SAMPLE
MULTIVARIATE LOCATION PROBLEMS















BY


SHOW-LI JAN


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


1991





























Copyright 1991

by

Show-Li Jan





























To my mother and to the memory of my father















ACKNOWLEDGEMENTS


I am very thankful to Dr. Ronald H. Randles, without whom this dissertation would not have been possible, for being my dissertation advisor. His encouragement, helpful comments, kindness, ideas, and invaluable assistance are important contributions to the development of this dissertation. Very special thanks are due Dr. Malay Ghosh, Dr. Pejaver V. Rao, Dr. Jane F. Pendergast, and Dr. Louis S. Block for serving on my committee and for many useful suggestions. I also thank Dr. Michael Conlon and Mr. Rodger E. Hendricks for many helpful comments on computing using fortran and IMSL subroutines. The continual support of my family, my parents-in-law, and my friends Ms. Taipau Chia and Mr. Shanshin Ton is acknowledged with appreciation.

Finally, most important of all, I would like to express deep gratitude to my husband, Gwowen, for his excellent typing and encouragement, without which this dissertation would not have been completed.










TABLE OF CONTENTS


ACKNOWLEDGEMENTS ......................................................................

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

ABSTRACT .......................................................................................

CHAPTERS


1

2


3






4 5









APPENDICES

A B


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

A MULTIVARIATE SIGN TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS ...... 14

2.1 Definition of the Test Statistic .......................................... 14
2.2 Null Distribution of Vn ................................................. 17
2.3 Asymptotic Distribution of Vn under Contiguous Alternatives ..... 19
2.4 The Pitman Asymptotic Relative Efficiency of Vn Relative to
H otelling's T2 ............................................................. 24

A MULTIVARIATE SIGNED-RANK TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS ...... 33

3.1 Definition of the Test Statistic ......................................... 33
3.2 A sym ptotics ............................................................... 34
3.3 Numerical Evaluation of ARE(nWn, T2) .............................. 41

MONTE CARLO STUDY .................................................... 47

A MULTIVARIATE SIGNED SUM TEST BASED ON INTERDIRECTIONS FOR THE ONE-SAMPLE LOCATION PRO BLEM ....................................................................... 66

5.1 Definition of the Test Statistic .......................................... 66
5.2 Some Intermediate Results ............................................. 70
5.3 Asymptotic Null Distribution of SS ................................... 77
5.4 Asymptotic Distribution of SS under Contiguous Alternatives ..... 86 5.5 Numerical Evaluation of ARE(SS/4t2, T2) .......................... 92



*��..�*�.... �. �... *... ��*.�....�..�.�.�.......................o�.�.o... ....... 98

..................................................................................... 105











BIBLIOGRAPHY ................................................................................. 107

BIOGRAPHICAL SKETCH .................................................................. III










LIST OF TABLES


2.1 A RE(Vn, T2) ....................................................................... 32

3.1 ARE(nWn, T2), WITH ARE(Vn, T2) IN PARENTHESES ................... 45

3.2 ERROR ESTIMATE ERREST OF ARE(nWn, T2) ........................... 46

4.1 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL
DISTRIBUTION WITH F, = 14 ................................................. 53

4.2 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL
DISTRIBUTION WITH F, = E ET .............................................. 54

4.3 MONTE CARLO RESULTS FOR ELLIPTICALLY SYMMETRIC
DISTRIBUTIONS WITH DENSITY OF THE FORM GIVEN IN
(2.3.1) W ITH Y = _4 ............................................................. 55

4.4 MONTE CARLO RESULTS FOR ELLIPTICALLY SYMMETRIC
DISTRIBUTIONS WITH DENSITY OF THE FORM GIVEN IN
(2.3.1) WITH = EET .....................................57

4.5 MONTE CARLO RESULTS FOR PEARSON TYPE II
DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY
OF THE FORM GIVEN IN (4.1.3) WITH Z = 14 ............................. 59

4.6 MONTE CARLO RESULTS FOR PEARSON TYPE II
DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY
OF THE FORM GIVEN IN (4.1.3) WITH = E ET .................60

4.7 MONTE CARLO RESULTS FOR PEARSON TYPE VII
DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY
OF THE FORM GIVEN IN (4.1.4) WITH F, = 14 ..............61

4.8 MONTE CARLO RESULTS FOR PEARSON TYPE VII
DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY
OF THE FORM GIVEN IN (4.1.4) WITH 2 = E ET .................62

4.9 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL
MIXTURES WITH MEAN 11 AND VARIANCE-COVARIANCE
MATRIX Z GIVEN IN (4.1.5) ................................................ 63

5.1 A RE(SS/4t2, T2) ................................................................... 96









5.2 ERROR ESTIMATE ERREST OF ARE(SS/4t2, T2) ............97














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
INTERDIRECTION TESTS FOR REPEATED MEASURES AND ONE-SAMPLE MULTIVARIATE LOCATION PROBLEMS

BY

SHOW-LI JAN

March, 1991

Chairman: Dr. Ronald H. Randles
Major Department: Statistics

Affine invariant interdirection tests are proposed for a repeated measures problem. The test statistics proposed are applications of the one-sample interdirection sign test and interdirection signed-rank test to a repeated measurement setting. The interdirection sign test has a small sample distribution-free property and includes the two-sided univariate sign test and Blumen's bivariate sign test as special cases. The interdirection signed-rank test includes the two-sided univariate Wilcoxon signed-rank test as a special case. The asymptotic null distributions of the proposed statistics are obtained for the class of elliptically symmetric distributions. In addition, the asymptotic distributions of the proposed statistics under certain contiguous alternatives are obtained for elliptically symmetric distributions with a certain density function form. Comparisons are made between the proposed statistics and several competitors via Pitman asymptotic relative efficiencies and Monte Carlo studies. The interdirection tests proposed appear to be robust. The sign test performs better than the other competitors when the underlying distribution is heavy-tailed or skewed. For normal to light-tailed distributions, the Hotelling's T2 and signed-rank test have good powers when the variance-covariance structure of the










underlying distribution is non H-type, otherwise ANOVA F and the rank transformation test RT perform better than the others.
An alternative test for the one-sample multivariate location problem is also proposed which extends the univariate signed-rank test to multivariate settings. The test proposed is somewhat like applying the interdirection sign test to the sums of pairs of observed vectors. It includes the two-sided univariate Wilcoxon signed-rank test as a special case. The asymptotic distributions of the proposed statistic under the null hypothesis and under certain contiguous alternatives are obtained for a class of elliptically symmetric distributions. Comparisons are made between the proposed statistic and Hotelling's T2 via Pitman asymptotic relative efficiencies. The signed sum test proposed performs better than Hotelling's T2 when the underlying distribution is heavy-tailed. However, for normal to light-tailed distributions, the Hotelling's T2 performs slightly better than the proposed test.














CHAPTER 1
INTRODUCTION


In this dissertation we investigate test statistics for certain repeated-measures and one-sample multivariate location problems. For the repeated-measures problem, we let Y1, ..., Yn be independently and identically distributed as Y, where Y = (Y1, ..., yp)T is from a p-dimensional, p > 2, absolutely continuous population. For each subject i, we shall regard Yi as repeated measures with one observation for each of the p treatments. We use the general mixed model with a subject by treatment interaction. In vector form, we consider the model (see, e.g., Winer [1971], p. 278), Y--i = ilp + + i + i, i =1...n, .(1.1) where lp is the pxl vector of I's, :j = (,c1, ..., tp)T is the vector of fixed treatment effects, and random variables Pi, Di, i for i = 1, ..., n are all mutually independent. (More details about this model will be given in section 2.1.) Note that the variance-covariance matrix of fti is probably not H-type, to be described later. We are concerned with the test of equal treatment effects, which in model (2.1.1) can be described as:

H0� , = T2 .. p versus Ha: j # -tj' for somej j. (1.1.2)


We first consider several parametric statistics for this problem when the population is p-variate normal. Probably the most well-known parametric procedure for this problem is based on Hotelling (1931) T2 test. Define the (p-1)-variate random vector Zi by

Z, = (Zil 7_.. Zi-)T = (Yil-Yip, ..... y, p-l-Yip)T, i = 1, ..... n.









The test of (1.1.2) can be carried out by the Hotelling's T2 statistic computed from the mean vector and sample variance-covariance matrix of the vectors of differences Zi's. This is due to Hsu (1938). The Hotelling's T2 is then defined as

T = T2( Z1, n -1(1.1.3)


where

nn
- n -" in (. )_ T
Z =-j--X~~- Zi,andE_. n-lil



Under H0, T2 is asymptotically Chi-square with p-1 degrees of freedom. If the underlying population is p-variate normal, then the null distribution of T2 is a multiple of the F-distribution with p-I numerator degrees of freedom and n-p+1 denominator degrees of freedom. Hotelling's T2 is invariant with respect to nonsingular linear transformations of the observations Zi, i = 1, ..., n. That is, if D is any nonsingular (p-1)x(p-1) matrix, then


T2M Z1, ..., D. ) = T2(Z1, ..., Zn ). (1.1.4)

We shall call this invariance property affine-invariance. This appealing invariance property ensures that the value of the test statistic remains unchanged following rotations of the observations about the origin, reflections of the observations about a (p-2)-dimensional hyperplane through the origin, or changes in scale. Hence the performance of Hotelling's T2 test will not depend on the structure of the population variance-covariance matrix or the direction of shift.
Another parametric procedure for this problem is the classic ANOVA F test. The test is based on the original observations Yij, I < i < n, 1 < j < p, and is defined as


n Y (YY)2
F -ji ,(1.1.5)
1 P
n-i . I I(Yij-Yi-Y -j+Y.)2
i=1 j=1









- in lP- lP
where Yj = n Y-j, Y Yj, and Yi. = - Yj. If the underlying population is
J n IJ I P j=l =

p-variate normal with variance-covariance matrix of the form



1P .. P
P *". P
- : p p , (1.1.6) p...p 1


then the null distribution of F is F-distribution with p- 1 numerator degrees of freedom and (n-1)(p-1) denominator degrees of freedom. This test for (1.1.2) under the variancecovariance matrix (1.1.6) was obtained by Wilks (1946) from the generalized likelihood ratio principle when the underlying population is p-variate normal. The matrix in (1.1.6) is said to have compound symmetry. While compound symmetry is a sufficient condition for the test statistic F to have an exact F-distribution, it is not a necessary one. Huynh and Feldt (1970) have found a necessary and sufficient condition, which may be expressed in three alternative forms (see Morrison [1976], p.152):
(1) The population variance-covariance matrix 1- (a..,), 1 < j, j' < p, has the pattern defined by
aj+axj,+X. if j j' 1..7 a{j+oj, if j j'



where X, cc1, ..., ap are (p+l) arbitrary constants such that the resultant matrix is positive definite.
(2) All possible differences Yj-Yj, (j j') of the response variates have the same variance.


(3) Define C, a function of the elements of 1, by









p 2(ad _ d..) 2
P P 2 2

(p-1) (j II a-, - 2p1 2 +p d2)
j=1 j'=_1 J j=1 i

where f&d= I= pj,6.= lj' and a. P=I '


Then an alternative statement of the necessary and sufficient condition is that C = 1. It may be noted that the scalar factor E is the multiplicative adjustment factor for the degrees of freedom proposed by Box (1954) and by Greenhouse and Geisser (1959). Any matrix whose elements satisfy (1.1.7) is referred to as a matrix of Type H. If the underlying population is p-variate normal with H-type variance-covariance matrix, the power of the F test will always be greater than that of the Hotelling's T2 test for the same alternative because the second degrees-of-freedom parameter of the F test is greater than that of Hotelling's T2 test. However, unlike Hotelling's T2, ANOVA F does not have affineinvariance. Thus, the performance of F test depends strongly on the structure of the variance-covariance matrix of the underlying population.

Morrison (1972) proposed several tests for this problem under various assumptions on the variance-covariance matrix. The tests include the usual Hotelling-Hsu statistic and the classical ANOVA F test as special cases.

Many nonparametric competitors to Hotelling's T2 have been proposed. A wellknown statistic is a rank test due to Friedman (1937). His test statistic is based entirely on within-block information and ignores between-block information. Let Rij denote the rank assigned to Yij within subject i. Let Rj represent the sum of the ranks associated with
n
treatment j, i.e., Rj = Y Rij, j = 1, ..., p. The Friedman test statistic for this case with no i=l

ties is

12 P 2
1 -np(p+l) R. 3n(p+l). (1.1.9) J=J









Under H0, the test S is asymptotically Chi-square with p- I degrees of freedom. (See, e.g., Hollander and Wolfe [1973], p. 140.) Define (n-i)S
= n(p-1)-S

Then under H0, the test FS is compared with an F-distribution with degrees of freeedom p-1 and (n-1)(p-1). A more accurate approximation to Friedman's test is proposed by Jensen (1977). Iman and Davenport (1980) proposed two new approximations and also pointed out that the F approximation is better than the X2 approximation.

Another rank test was proposed by Koch (1969). His test statistic used the ranks of the aligned observations, obtained by subtracting from each observation the average of the observations for that block. This alignment process will eliminate or at least reduce the block effect. To introduce his test, let us define

Rij = rank of (Yij-Yi.) among Yl1-Y1., ...,Ynp-Yn.,


- In - IP Rj = - Y, Rij and Ri. =- 7 Rij.
n i__1 Pj=1


Then the test statistic has the form P nL+l 12
nX (R.j - 2
W= _ j~ (1.1.10)
1
n p(Rij - Ri.)
n _-1 j=l

If 11, ... Yn are i.i.d., then under the null hypothesis of the exchangeability of the components of Yi = (Yil, ...,Yip)T, the test W* has an asymptotic Chi-square distribution with degrees of freedom p- 1.

Unlike the Friedman test, which depends entirely on within-block rankings, Quade (1979) considered a p-sample extension of the Wilcoxon signed rank test by taking









advantage of the between-block information. This is done by considering weights assigned to each block on the basis of some measure of within-block sample variation, such as the range, standard deviation, mean deviation, or interquartile difference. To illustrate this test statistic, let us define

Di = D.Y), a location-free statistic that measures the variability within the ith block,


Qi = rank of Di among D 1, ..., Dn, and


Rij = rank of Yij among Y I,..., Yip.


Quade's procedure, based on weighted within-block ranks, is defined as p n ]
72 [ QiRj]2
W j= Il i il 9(p+l)n(n+l)
p(p+l)n(n+l)(2n+l) 2(2n+l) p n . I]
721 1 Qi(Ri- "2
=1- i=l 11.1
p(p+l)n(n+l)(2n+l)

Under the null hypothesis of the exchangeability of the components of Y., the test W has an asymptotic Chi-square distribution with p-1 degrees of freedom.
A natural nonparametric analog of ANOVA F test was proposed by Iman, Hora and Conover (1984). Their procedure is first to transform all the observations to ranks from 1 to np and then apply the parametric ANOVA F test to the ranks. This approach retains both the within- and between-block information. Defining Rij = rank of Yij among YI1, .... Ynp,


the rank transformation test is defined as









P

RT = 1 l - (1.1.12) n---' 7,' Yl (Rij-Ri'-PR'J+R.")
1=1 J=1

1n 1P 1P
where Rj= Rij, I Rii and R.. - P R.j. Assuming Yj's are mutually
n=M P j=1 p j1

independent, and considering

H0 : Fij = Fi for i = 1, .... n and j = 1, ..., p, where Fij is the distribution function of Yij, then the asymptotic null distribution of (p-l)RT is Chi-square with p-I degrees of freedom under suitable conditions. Their simulations showed that the behavior of test RT is closely approximated by the F-distribution with (p-1) and (n-1)(p-1) degrees of freedom. Comparisons made among ANOVA F test, Friedman's test (S), Quade's test (W), and rank transformation test (RT) via Monte Carlo studies showed that the F test had the most power for normal distributions, the Quade's test and F test were almost equivalent and gave the best results for uniform distribution, the Friedman's test and the RT test gave similar results and were best for the Cauchy distribution, and the RT test has the most power for double exponential and lognormal distributions. Hora and Iman (1988) developed the limiting noncentrality parameters of the rank transformation statistic and some other tests, which were then evaluated to make comparisons among those tests via Pitman asymptotic relative efficiencies.

Agresti and Pendergast (1986) also considered a test that is appropriate when the null hypothesis (1.1.2) is expressed as the exchangeability of the components of Y-i. Their procedure utilizes a single ranking of the entire sample. Let

- n
Rij = rank of Yij among YI, ... Ynp, jn Rip ji=I









p = Corr(Rij, RiP,), j j', and 2 = Var(Rij).


The test statistic is based on

n ( (R p)2
T= j=1 2(1.1.13)


This test includes Koch's test and the rank transformation test as special cases. They argued that under H0 their statistic has an asymptotic Chi-square distribution with p-I degrees of freedom if the asymptotic distribution of R = (R. 1, ..., R.p_ 1)T is (p-i )-variate normal. Since they did not present conditions guaranteeing this normality, Kepner and Robinson (1988) concluded the work of Agresti and Pendergast by determining reasonably sufficient conditions for R to have a (p-1)-variate normal limiting distribution.

In analogy to the Iman, Hora and Conover (1984) proposal of a rank transformed version of ANOVA F test, Agresti and Pendergast (1986) considered a rank transformed version of Hotelling's test. This procedure is appropriate when the hypothesis of no treatment effects is more broadly expressed as the marginal homogeneity condition F1 = F2 = ... = Fp, where F1, ..., Fp denote the one-dimensional marginal distribution of Y = (Y1, yp)T. Their statistic is based on


RTH = nT'., (1.1.14)


where
in _ n
-.R .., RTand T n I Ri' Ri = (Ri I i P'I-RP) 'and a = n - 1 1 (i L)
1=_ i=1

Their simulations showed treating the null distribution of RTH as a multiple of Fdistribution with p-I numerator degrees of freedom and n-p+ 1 denominator degrees of freedom is a reasonable approximation. They also argued that under H0, their statistic has









an asymptotically Chi-square distribution with p-1 degrees of freedom if the asymptotic distribution of (R.1, .... 1l)T is (p-1)-variate normal. Their simulations showed that the RT and RTH statistics behaved much like their parametric analogs.

We now consider the multivariate tests for the one-sample location problem. As a first step, we let X1, ..., Xn be i.i.d. as X = (X1, ..., Xp)T, where X is from a p-variate absolutely continuous population with location parameter e (pxl). We would like to test HO:e =0 versus Ha:0 0. (1.1.15) Here 0 is used without loss of generality, since H0� = % can be tested by subtracting %.0 from each observation Xi and testing whether these differences (Xi - %)'s are located at O.
The classical procedure used in this setting is Hotelling's T2, which is defined as T2 = T2( X1, .,X n X T..'x,


where

in in
X-- Xi, and Z= n-i il


If the underlying population is p-variate normal, then the null distribution of T2 is a multiple of the F-distribution with p numerator degrees of freedom and n-p denominator degrees of freedom. The affine-invariant property of Hotelling's T2 is discussed earlier on page 2.
Many nonparametric procedures have been proposed. The most popular statistic is the component sign test, which is a nonparametric analog of Hotelling's T2 using signs of Xij's, 1
n
Sj =Y sgn(Xij), and S = (S1, ..., SP)T,
i=l









where sgn(x) = 1(0, -1) for x >(=, <) 0. The test is based on S* =T (1.1.16) 1in

where= (wjj), wsgn(xij)sgn(xij) for 1 < j, j' < p. Under H0, the test statistic Sn is asymptotically Chi-square with p degrees of freedom. Sen and Puri (1967) considered other score function versions of this statistic as well. Bennett (1962) considered a similar sign test for the problem of testing the equality of location parameters in two p-variate (p < 4) populations. Chatterjee (1966) also studied a similar procedure for a bivariate case. In the paper of Bickel (1965), he proposed several general test statistics based on Hotelling's T2. They are Hotelling's tests of type I, M 2 and W 2, and Hotelling's tests of type II, 4 2 and 0 2. His p-variate test statistics M 2 and '0 2, and W 2 and a,2 are quadratic forms involving coordinate-wise sign and Wilcoxon signed rank statistics, respectively.

Unlike Hotelling's T2, the above competitors to T2 are not invariant under nonsingular linear transformations of the observations. Consequently, their power and efficiency depend on the direction of shift and the variance-covariance matrix of the alternative distribution. In an effort to overcome this problem, Dietz (1982) proposed sign and signed rank tests for bivariate location settings. Her tests are three-step procedures. First, a certain transformation is applied to the observations. Second, new coordinate axes are chosen. Third, standard sign and signed rank tests are performed using the transformed and rotated observations. In Dietz (1984), she extended those results to general linear signed rank tests for multivariate location problems. Here, however, a two-stage procedure is recommended, corresponding to steps one and three of the bivariate case. The resulting test statistics are no longer invariant under linear transformations, but for elliptically symmetric alternatives, their asymptotic efficiencies are independent of the direction of shift and the variance-covariance matrix. Her simulation study showed that linear









transformations have little effect on the small sample power of the tests for nearly degenerate distributions.
Two well-known bivariate sign tests are due to Hodges (1955) and Blumen (1958). Their procedures are affime-invariant and have a distribution-free property. Joffe and Klotz (1962) presented an expression for the exact null distribution of the Hodges bivariate sign test. They also computed the Bahadur limiting efficiency of the test relative to the Hotelling's T2 test for normal alternatives. Killeen and Hettmansperger (1972) made an exact Bahadur efficiency comparison of Hotelling's T2 with respect to both Hodges' and Blumen's bivariate sign tests. Klotz (1964) obtained exact power for the bivariate sign tests of Hodges and Blumen under normal alternatives and therefore permitted comparisons of the two tests for sample sizes n = 8 through 12.

The procedure proposed by Bennett (1964) for the bivariate case is a signed-rank test generalizing Wilcoxon's univariate signed-rank test. This test is not affine-invariant.

Another affine-invariant bivariate rank test was introduced by Brown and Hettmansperger (1985). Their statistic is based on the gradient of Oja's measure of scatter (Oja, 1983). Letting A(Xi, Xj; 0*) denote the area of the triangle formed with Xi, Xj and a* as vertices, define

T(9*) = Z YA(Xi, Xj; Q9*). (1.1.17) i
This is the Oja measure of scatter. The value A* which minimizes Tj*) is the Oja generalized median of the bivariate sample. Brown and Hettmansperger proposed the use of QnQ*) = DT(Q*), the vector of partial derivatives of T(*). The generalized median of Oja is the value which minimizes I Qn(*) I. Define n n T
S= ZQ2n(-Xi), and C= I Q2n(-Xi)Qn(X), (1.1.18) i=l i=1









where Q2n(-Xi) is computed using observations X1, ..., Xn, and their projections through the origin, -X1, ..., -Xn. Their test for (1.1.15) is defined as STc'IS. Under H0, the test statistic is asymptotically Chi-square with 2 degrees of freedom.

Oja and Nyblom (1989) also studied the bivariate location problem. Their tests are analogs of the univariate sign test. Denote the direction angle of Xi by Oi. Then 0i = Oi+n (mod 2n) is the direction angle of -Xi. Write 01 < 02 < ... < 02n for the ordered angles in the set 0, ...On, 0 1, ..., n). Define


Zj= 1 if0ie {01, ...,0}


0 if0iE [0*, 0*), i = 1, ..., n, (1.1.19)


and

Zn+i = 1-Zi, i = 1, 2, (1.1.20) The vector Z = (Z1, ..., Zn)T indicates which of the observations lie above or below the horizontal axis. They proposed using test statistics based on Z. The test statistics are distribution-free and affine-invariant, and include Hodges (1955) and Blumen (1958) sign tests as special cases. Also, they proposed some new intuitively appealing tests. A general class of these invariant sign test statistics is n-1I n ~ 2
I I. (Zk+-').h(i/n)] (1.1.21) k=0 i_-1


where h is a suitably chosen score function.

In the recent study Randles (1989) proposed an interdirection sign test for this problem. His test statistic included the two-sided univariate sign test and the Blumen (1958) bivariate sign test as special cases. Also, Peters and Randles (1990) suggested a signed-rank test based on interdirections, which includes the two-sided Wilcoxon signed-









rank test as a special case. The bivariate case of their statistic was considered in the dissertation of Peters (1988). The interdirection sign test and the interdirection signed-rank test will be described in detail in Chapters 2 and 3, respectively, where they are applied to a repeated measures problem.
In this dissertation, the interdirection sign test for a repeated measures problem is defined in Chapter 2. The asymptotic distributions of the test under H0 and under certain contiguous alternatives are obtained in sections 2.2 and 2.3, respectively. The Pitman asymptotic relative efficiencies of the test relative to Hotelling's T2 are presented in the last section. In Chapter 3, the interdirection signed-rank test for the same problem is described, with its asymptotic distributions obtained in section 3.2 and the evaluations of the ARE of the signed-rank test relative to Hotelling's T2 established in the section 3.3. Comparisons of several competing procedures are made in Chapter 4 via Monte Carlo studies. An alternative test for the one-sample multivariate location problem is proposed in Chapter 5. Some useful intermediate results are presented in section 5.2. The asymptotic distributions of the test under Ho and under certain contiguous alternatives are developed in sections 5.3 and 5.4, respectively. Finally, in section 5.5, we evaluate the ARE of the proposed test relative to Hotelling's T2.














CHAPTER 2
A MULTIVARIATE SIGN TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS



2.1 Definition of the Test Statistic

The multivariate sign test based on interdirections, denoted by V., was proposed by Randles for the one-sample multivariate location problem. In this section we will show how this test statistic can also be applied to repeated-measures designs for detecting treatment effects.
For the one-sample multivariate location problem, we let X1, ..., Xn, where Xi = (X,1, ... Xip)T, be independent and identically distributed (i.i.d.) as X_ = (X1, ..., XP)T, where X is from a p-dimensional absolutely continuous population with location parameter

0 (px 1). We would like to test

H0"0.=0 versus Ha:a* #0.

Here 0 is used without loss of generality, since H0 :* = -0 can be tested by subtracting 1% from each observation Xi and testing whether these differences (Xi - .0)'s are located at

0..
For the problem of single-factor repeated-measures designs, we let Y1, ..., Yn be i.i.d. as Y = (Y1, ..., Yp)T, where Y is from a p-dimensional, p > 2, absolutely continuous population. Note that the components of Yi are repeated measurements of the ith experimental unit. We will use the general mixed model with a subject by treatment interaction. In vector forms, we consider the model (see, e.g., Winer [1971], p. 278), Y---i = ilp +.I + i + i, i = 1, ..., n, (2.1.1)






15


where lp is the pxl vector of l's, I = (@1, ..., rp)T is the vector of fixed treatment effects, Pi represents the random effect of the ith subject, Di denotes the vector of the ith subject by treatments interactions, and Fi is the vector of random error of the ith subject. We assume 3i's are i.i.d. with mean 0, j.i's are i.i.d. with mean 0 and a general variancecovariance matrix possibly not H-type, described in Chapter 1, i's are i.i.d. with mean 0
2
and variance-covariance matrix ae.Ip, where Ip is the pxp identity matrix. The random variables 3i, li, E for i = 1, ..., n are all mutually independent. We are concerned with the test of equal treatment effects, which in model (2.1.1) can be described as:

H0: tI =T2 =...tp versus Ha: ti 'rj' for somej #j. (2.1.2) Note that the sample Y1 ..., Yn has location parameter t. The problem is to test whether the components of the location parameter 1. are all equal. We can transform this problem to the standard one-sample multivariate location problem, described earlier, by looking at the differences among the components Yij within each observation Yi. The transformation is described below. Define

Yil-Yip Yil Yi2-YiP [1 ! Yi2 Zi= = Ai ,=1..,n,

1 _ Yi'p-1
1-Yi p-l-Yip-i L Yipand

- t 1.p "C1 ,l;2--Tp T12

..p� =Atp
-Xp - -Cp -Tp
-Tp-










where


1

A = (2.1.3)



is a (p-l)xp matrix. Now, we have a (p-1)-variate sample Z1, ... Zn, which can be modeled via
Zi =0 + l ~*+f i = 1, ..... n,



where Q ((p-1)x 1) is the location parameter. Note that the variance-covariance matrix of
is possibly not H-type. Thus, testing the hypotheses given in (2.1.2) is equivalent to testing

H0� 0= versus Ha: Q0 # . (2.1.4) We have shown that the test of (2.1.2) based on p-variate observations Y1, ..., Yn can be carried out by using a multivariate location test based on a statistic like the interdirection sign statistic Vn, computed on the transformed (p-1)-variate observations Z, ..., Zn, for testing (2.1.4). Now we will describe the test that rejects H0� 0 = 0 for large values of the statistic
P1 n rn (2..5 Vn =_n. I-, I cos~tPik), 215 ni=1 k=1


where

_ Ck+dn ifi;k Pik (pn)


0 ifi=k,


(2.1.6)









p- n-2


and Cik denotes the number of hyperplanes formed by the origin 0 and other p-2 observations (excluding Zi and Zk) such that Zi and 4 are on opposite sides of the hyperplane formed. The counts (Cik I 1 < i < k < n), called interdirections, are used via 7rPik to measure the angular distance between Zi and Z. relative to the positions of the other observations. This statistic, Vn, includes Blumen's bivariate sign test and the 2-sided univariate sign test as special cases. Also, it is affine-invariant and has a distribution-free property under H0, for a broad class of distributions, called distributions with elliptical directions, which includes all elliptically symmetric populations and many skewed populations as well. In the next section, we will concentrate on the family of elliptically symmetric populations.



2.2 Null Distribution of Vn

In this section we will find the null distribution of Vn under the class of elliptically symmetric distributions, which is defined below.


Definition 2.2.1 Assuming the existence of a density function, the mxl random vector X is said to have an elliptically symmetric distribution with parameters U (mxl) and I (mxm) if its density function is of the form

fX x = Kmll-1/2h[(. - UT1-I( - Ut)], (2.2.2) for some non-negative real-valued function h, where I is positive definite and Km is a scalar. We will write this distribution of X as Em(, ).









Throughout this chapter, we will use YI, ..., iYn to denote the original sample, and use Z1, ..., Zn, defined in (2.1.3), to denote the transformed sample, to which the test statistic Vn is applied. Now, let's assume Y1, ..., Yn are i.i.d. as Y = (YI, ... yp), where Y is Ep(a, j). To apply the result of null distribution of Vn under elliptically symmetric distributions, proved by Randles, we shall first prove that the transformed sample Z1, ..., Zn is also elliptically symmetric. To do this, we use the following lemma, which was given as an exercise in Muirhead's (1982) book.


Lemma 2.2.3 If X is Em(U, Z) then :

(i) the characteristic function (2X() = E(eitTX) has the form


-X() = eiil" V(.TJ 1 ) for some function v, (2.2.4) and
(ii) provided they exist, E(X) = l, and Cov(_) = cZY for some constant a.


Theorem 2.2.5 If Y is Ep(9-, ) and Z = A Y, defined in (2.1.3), then Z is Ep I(CA.g, A Y AT).

Proof of Theorem 2.2.5 Since Y is Ep("l, j_), by Lemma 2.2.3, the characteristic function of Y at 1, a px 1 vector, has the form of 1(1) = eilTA Vl(tT.t) for some function v. Thus, the characteristic function of Z at s, a (p-1)xl vector, is

E is _ isTAY
O -(a) E(eisTZ) = E(eis--)


= E(ei(Ts)T)

i[eATATT









= [eisT(A1)].W[ sT(AAT)s]. Thus, Z is E,.I(All, AEAT).


We are now prepared to state the following theorem.


Theorem 2.2.6 Assume the observations Y1, ..., Yn are i.i.d. from a p-variate elliptically symmetric distribution. Then, under H0, defined in (2.1.4), Vn, defined in (2.1.5), computed on observations Z1 ... n has a small-sample distribution-free property and a limiting 4 distribution.

Proof of Theorem 2.2.6 See Randles (1989), p. 1046-1047.



2.3 Asymptotic Distribution of Vn under Contiguous Alternatives

In this section we will find the asymptotic distribution of Vn under a sequence of alternatives approaching the null hypothesis H0� : = 0. In doing this, we will restrict our attention to a specific class of elliptically symmetric distributions. Let's assume Y1, ..., Yn are i.i.d. as Y = (Y1, ..., yp)T, where Y is elliptically symmetric with a density function fy of the form

fy(y) = Kpll-1/2exp[( ( .y _)/c0]V}, y e RP, (2.3.1) where

Sp(p/2v) p= vF(p/2) CO =[(p+2)/2v] =r'(p/2v)(Co)P/2' (2.3.2)



F(w) = JxWle-x dx for w > 0,
0










and RP is the Euclidean p-space. It can be verified that the expression in (2.3.1) is a valid density function and that U represents the mean and _, the variance-covariance matrix. This family includes the multivariate normal distribution (v = 1), heavier-tailed distributions (0 < v < 1) and lighter-tailed distributions (v > 1). As explained in the previous sections, we will need to derive the density function of the transformed sample, whose form will be used when deriving the Pitman asymptotic relative efficiency of Vn relative to Hotelling's T2 in next section.


Lemma 2.3.3 For the family of distributions given in expression (2.3.1), the transformed sample Z1, ..., Zn has a density function of the form

f = Kp IAT'Iz], _A e Rp-l, (2.3.4)


where

2
g(t) = f exp{-[(t+s )/Co]vlds. (2.3.5)
-00


Proof of Lemma 2.3.3 Letting x = -, then we can write


Y-Yp 1

LA

- Yp


where B = TI, p = (0, ..., 0, 1)T, a pxl vector with 1 on the pth component and 0's elsewhere, and A is defined in (2.1.3). Since B is nonsingular, it follows that Y =B'Ix. The jacobian of the transformation, denoted by J(y =* x_), is equal to 1E = 1BI"1 =1. Hence the density function of X is








f x-) = fy(Y-) I y- = B-lx

= KPIIP1/2exp (-[(_ lx-)T'l(B'lx-j)/CO]v},

= KpII'l/2exp {-[(X-B T(_Be -_.T)/C0]v . Since IB E BTI = II due to IBI= 1, it follows that

fx(x) = KlBXBTI/2exp ([(-B ( Letting B BT = V and B U = ., we rewrite the above expression as

fx( x) = pV-/2exp -[x-_V} Thus, the density function of Z is

fz ) = f fx(.) dyp
-00

= KpIVI"1/ exp{-[( - )TVdl x- )/c0]VIdyp. (2.3.6)
-00

Note that

V = BeBT = &T A p and -- = B U [A A gpYA P- p pL 1 FV1l Y12 T
Denoting V by V21 V22 j, we have V1 = A A is nonsingular and V22 =- g P is a positive scalar. Next we use the fact that

V-1 _ [ Y 1 M12 ]1
-- Y21 V22









-1 -1 1 Yll11 + Vll-YV 2.1VY21Vll


Vi2 .1


where

V22.1 = V22 - y21V-11V12. We can expand the quadratic form in expression (2.3.6) as

(x-nTv 1 (x-n)
TV'I V -1 -1
Y l TVl(z-) + z _1 2V22.121 -_)


(YP-I) 2 V1 - 2(Q)T V-12 -1 AP-)





Using the fact that IVI = VIII.I V22- V21Vl11VI21= V111.IV22.11 and expression (2.3.7), we may write expression (2.3.6) as


fz)

= KplV11"l/2.1V22.11-1/2


00
fO Vxf[(Z -1/2 T -1 -1/'2 )2/OVId
0 exp{-[((- -)T-Vll(Z-) + ((YP-I'P)V22.1 - ( -) 11---12V22.1)
-00

Takings= (yp-p) - 1/2 T w1 -1/2
Takng =(ypgpV22.1 - (7--Q) V11V2V22.j, we can write the above expression as


(2.3.7)


-1_ - I
-22.1V-21V 11








00
fz. = KpIVIII"'/2 f exp{-[((z-_)TV (z-_) + s2)/c0]V ds. (2.3.8)
-00

Since V = A rAT and Q = Ay1, expression (2.3.8) is equivalent to expression (2.3.4). This completes the proof. It can be verified that expression (2.3.8) is valid density function.


Now, we are in the position to discuss the asymptotic distribution of Vn under a sequence of alternatives approaching the null distribution. Under H0, Z1, ..., Zn are i.i.d. with density of the form

f(z) = KIA ZATrl/2 f exp{-[(zT(--AT)lz+s2)/c0]V}ds. (2.3.9)


Under a sequence of alternatives let Z1, ..., Zn be i.i.d. with density f (z-_n-I/2 ), where fZ is given in (2.3.9) and c e RP-'-{0} is arbitrary, but fixed. It is shown, with the outline of the proof given in Appendix A, that if 4v + p > 3, then
0 < k~Z) = afz(-Z ]2 f
0

Thus, when 4v + p > 3, the rationale of Hdjek and Siddk (1967), p. 212-213 and earlier, shows that the alternatives are contiguous to the null hypothesis. Noting that both Vn and Hotelling's T2 are affine-invariant, we can, without loss of generality, assume throughout thatAlAT = !p-l, the (p-l)x(p-1) identity matrix.


Theorem 2.3.10 Let Y1, ..., in be i.i.d. with density function fy given in expression (2.3.1). Assume the density function of Z1, ..., Zn has the form of
0o
fZ) = Kpg( z) = Kp f exp{-[(z'z I)C0]V}ds, z r RP'.
-00









If 4v + p > 3, then under the sequence of contiguous alternatives, d2 ( {E [Rg(R)] }cc), (2.3.11) Vf P-. P-1 HO g(R2)

where
00
g(R2) f j exp{'[(R2+s2)/Co]V)ds,
-00


and

g'(R2) = dg(R2)/dR2 and R2 =zTZ.

Proof of Theorem 2.3.10 See Randles (1989), p. 1050.


Note that the noncentrality parameter in (2.3.11) can be simplified and this will be done in the next section.



2.4 The Pitman Asymptotic Relative Efficiency of Vn Relative to Hotelling's T2


In this section we will use Pitman relative efficiencies to make comparisons between V. and Hotelling's T2in the repeated measurement design settings. Because these statistics are all affine-invariant we may, without loss of generality, make the simplifying assumption that the transformed sample variance-covariance matrix is the identity. We will apply the asymptotic results under the contiguous alternatives in Theorem 2.3.10.

The Pitman approach to asymptotic relative efficiency compares two test sequences {Sni} and {Tmi} as the sequence of alternatives Hi� :0 = _i, approaches the null, which we are taken to be H0: = 0. The subscripts ni and mi are the sample sizes for tests Sni and Tmi, respectively. Let 13 {Sni, 0Qi} and 1I Tmi, 0i} denote the powers of the tests based on









Sni and Tmi, respectively, when 0 = 0i. Assume that ni and mi are such that the two sequences of tests have the same limiting significance level Cx and cc< Um P {Sni, .lim f[{Tmi, j} < 1.
1--)00 1 --Then the Pitman asymptotic relative efficiency (ARE) of { SnI relative to {Tmi) (or simply of S relative to T) is

ARE(S, T) = lim
1 )0 ni

provided the limit exists and is the same for all such sequences {ni) and (mi), and independent of the (i) sequence. (See, e.g., Randles and Wolfe [1979], p. 144.)

Hannan (1956) shows that if, under the sequence of alternatives Hi, the test sequences (Sni) and (Tmi} are asymptotically noncentral Chi-square with the same
2 2
degrees of freedom and noncentrality parameters, 82 and 8T, respectively, then


82
ARE(S, T) -2
aT

It's well known that under the sequence of contiguous alternatives described in the last section, and taking A ,AT = 1p-l, the asymptotic distribution of Hotelling's T2, where
n -.. 1
- nT ('IZ with_ = n1 YZi and E- (n-l)"1 _(ZiZ)(Zi-Z)T, is noncentral i=l i=l chi-square with p-1 degrees of freedom and noncentrality parameter
2 T
8 2= . (2.4.1)
T

(See, e.g., Puri and Sen [1971], p. 173.) To derive ARE(Vn, T2) we use the following lemma.









Lemma 2.4.2 Let Y1, ..., Yn be i.i.d. with density function given in expression (2.3.1), and R2 = zTZ. Taking Al AT = Ip.1, then, under H0: = 0, the density function of R2 is of the form


fR2(r) = Cp r(P')/2"g(r), r > 0, (2.4.3) rl[(p-1)/2]

where

g(r) = jexp{-[(r+s2)/Co]V}ds, (2.4.4)
-00

and Kp and CO are defined in (2.3.2). Proof of Lemma 2.4.2 Taking A I AT = Ip.1 and under H0: = -O, we have Z is E.p(Q, I.1) with density function of the form
00
fz.) = Kpg(zTz =K exp{-[(z-+s2)Ic0]V}ds,'z Rp'I
-00

Thus, R2 = ZTZ has density of the form given in expression (2.4.3). (See, e.g., Muirhead [1982], p. 37.)


Theorem 2.4.5 Assume Y, ... Yn are i.i.d. from a density given by (2.3.1). In the

repeated measurement settings, taking A AT = Ip.1, if 4v + p > 3, then the Pitman asymptotic relative efficiency of Vn relative to Hotelling's T2 is


ARE(Vn' T2) = 414(p/2)F2[(p- 1)/2v]F[(p+2)/2v]
p(p-1)F4[(p-l)/2]F'3(p/2v) , p > 2. (2.4.6) Proof of Theorem 2.4.5 If 4v + p > 3, taking AAT = Ip.1 and using expression (2.3.11), we have, under the sequence of contiguous alternatives, the test Vn is









asymptotically noncentral chi-square with p-1 degrees of freedom and noncentrality parameter ,n' where


2~, 4A~{E FRg'(R2)2 T Vn = P H g(R2)


It follows that 622

ARE(Vn, T2) - - 0 g(R 2 (2.4.7) 82 2 - Og(R2)


Under H0, we find, using expressions (2.4.3) and (2.4.4),


R.g'(R2)


g(R2)

** '4rg'(r)
47 g(r) fR2(r)dr

Jo lrg'(r) Kp75P')/2
P r( - 1)/2-1 g(r)dr
g(r) ]F((p-1)/2)

K p 7(P-1)/2

= \(F'((p-1)/2) rpi g'(r)dr.
0

It can be verified that

g'(r) = ( f exp { -[(r+s2)/CO] V) ds)'
-0o

= 00 [(exp{-[(r+s2)/Co]V}) ] ds.
f 17 s CO a1









(See, e.g., Trench [1978], Theorem 5.2 and 5.6 on p. 581 and p. 586, respectively.) Letting


Kp (p-1)/2
Cp=F[(p-1)/2]'


(2.4.8)


we have


ERgI(R2)]
g(R2)


= Cp 7rp/2-1 0[�a(expf-[(r+s 2)/C0]v})] dsdr
0 -*"r

= Cp jrr/2- exp{-[(r+s2)/C0]V)(-v)((r+s2)/C0)v'l CI ds dr
00


" Co)


0000
fJ frp/2-1 exp{f [(r+s2 )/CO]V} (r+s 2 v-i ds dr~.


Taking r = t2, we can write


E[ Rg'(R2) ]
g(R2)


4 C ) 00 f _ [(t2+s2)/c] V}(t2+s2)V I .
0j tP' exp{-~ s)CJ}t+ ds dt. (2.4.9)


Letting s = - 0 xl/Vsin(0) and t = 4C xl/Vcos(0), we find that

as/a0 = i JC xl/Vcos(O), as/Dx = v4-Co x(lv)/Vsin(O), at/a0 = -4JC xl/vsin(O), at/ax = vl-4J x(1-v)/Vcos(0),


s2+t = CO x2/v.









The jacobian of the transformation is

v-ICO x(2-v)Ncos2(0) + v'IC0 x(2-v)/Vsin2(o) = V-1C0 x(2-v)A'. Thus, we can write (2.4.9) as
E[R'g'(R2)
ELg(R2) ]

(-P o7r/2 x/Vcos(0))P'l(Cox2/V)Vlexp('x2)v'lc x(2v)' dOdx
co o 6li!o o (0)) (C 'C


(.C~(-1)/2) (0 (+V-l)/v 2 ne/2p= " . x lexp(-x )dx).( " cosP (O)dO).
0 0

The constant term in the product above is
4CpC(PI)/2


Kpg(p" 1)I2 C(_1/
-4 - 1)/2 C~'1l)2 (using expression (2.4.8))
F((p-1)/2)


(2.4.10)


-4 v (p/2)
r(p/2v)(icCo)P/2


-4 vf(p/2)
F(p/2v)F[(p-l)/21


(p- 1)/2 _(p. 1)/2 (using expression (2.3.2)) F[(p-1)/2] 0


1


The first integral in expression (2.4.10) is

00
f x(P+v'l)/Vexp(-x2)dx
0
1 00(p)/2V
2 fy y'P )2exp(-y)dy (using y = x2)
0









1 00 (p+2v-1)/2v-l 2P f Y V41Vexp(-y)dy
0


= rLF( p+2v- )
2 2v

1 (p-i) r(P-1
2 2v 2v


_1 r(P- +1)
2 2v

S(p i) r(p-1)
4v 2v


The second integral in (2.4.10) is

rt/2 n /2 2, j cosV"1(0)dO = f sinV'1(0)dO 2
0 0 2r(-P+)
2

(See, e.g., Beyer [1987], p. 289.) Thus, the expected value in expression (2.4.10) is

E R'g'(R2) ]
g(R2)


:[-4 vr(p/2) 1
r-(p/2v)r((p-1)/2) -If -0W


4v


] I


-(p-1) r2e(12L) fe( k)
2 2v
2r 2I


(2.4.11)


Therefore, using expressions (2.4.7) and (2.4.11), we have


2v- 21 2


(pl) r4(--.) r2 .PfI)
2 2v"l
F2(.-- F 2 .- F 2 ._L ,


[F( 2(-v) ]. (using (2.3.2)) PF(2-)


Noting that


ARE(Vn, T2) =









F(P+) -1 )[r-4)l ).


we have
-4P 2(_-2-i) F- p+2 (p-l1) r4@ 1-2, F 2-jk2 ARE(Vn, T2) - F(-P) 2v
pIF 32v k 2(@ 2-2


F4( P- 2, P-I p2 4 1r+ I"2( 2v1) I" 2--v2

pp-) F3 (v-v)F14 (-'2-1





This ARE is evaluted in Table 2.1 for selected values of v and p satisfying the condition 4v + p > 3. Note that when p 3, 4v + p > 3 for all v >0, and when p =2, 4v + p >3 for all v > .25. When the underlying population is multivariate normal or close to normal (v = 1.0, .75, respectively), Hotelling's T2 performs better, yet Vn appears to be quite competitive and the efficiencies increase as p increases. For light-tailed distributions (v = 2, 3, 4, and 5) the sign test is not as effective as T2, but the ARE's increase with p. For heavy-tailed distributions (V = .50, .25, .20,.15, and .10) Vn is more effective than T2. Here the efficiencies decrease with p. In fact, it can be verified, using Stirling's formula for approximation of m! and the result lirn (l+kx)l/x = eX for constant X, X-*oo

that for fixed v, ARE --- 1 as p -- oo. For fixed p, it can be shown that as v -- 0, the ARE -4 oo and as v--+ oo, the ARE ---> {4p2I"(p/2) )/{ (p+2)(p-1)3F4((p- 1)/2)}, the latter is evaluated for p = 2 to 12 and is displayed in Table 2.1 under the column v = oo.













Table 2.1
ARE (Vn, T2)

V

p oo 5.0 4.0 3.0 2.0 1.0 .75 .50 .25 .20 .15 .10

2 0.4053 0.4205 0.4278 0.4422 0.4784 0.6366 0.7851 1.2159 4.7283* 9.4001* 29.6622* 297.1379*
3 0.5552 0.5738 0.5822 0.5983 0.6364 0.7854 0.9110 1.2337 3.1089 4.9469 10.7403 50.7101
4 0.6404 0.6599 0.6682 0.6837 0.7191 0.8488 0.9519 1.2008 2.4262 3.4521 6.2165 20.1730
5 0.6971 0.7164 0.7203 0.7387 0.7708 0.8836 0.9700 1.1711 2.0674 2.7484 4.4185 11.4238
6 0.7378 0.7566 0.7640 0.7773 0.8063 0.9054 0.9796 1.1477 1.8496 2.3487 3.4980 7.7606
7 0.7687 0.7868 0.7937 0.8059 0.8323 0.9204 0.9852 1.1295 1.7043 2.0939 2.9512 5.8638
8 0.7930 0.8103 0.8167 0.8280 0.8521 0.9313 0.9888 1.1151 1.6008 1.9182 2.5934 4.7408
9 0.8126 0.8292 0.8351 0.8456 0.8677 0.9396 0.9912 1.1035 1.5235 1.7903 2.3428 4.0123
10 0.8287 0.8446 0.8502 0.8600 0.8804 0.9461 0.9929 1.0939 1.4636 1.6931 2.1583 3.5075
11 0.8423 0.8575 0.8628 0.8719 0.8908 0.9513 0.9942 1.0860 1.4160 1.6169 2.0173 3.1399
12 0.8539 0.8685 0.8734 0.8819 0.8996 0.9556 0.9951 1.0793 1.3771 1.5557 1.9061 2.8619


*Conjectured ARE's when p = 2 and v < .25, using the form of the efficiency given in Theorem 2.4.5.














CHAPTER 3
A MULTIVARIATE SIGNED-RANK TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS



3.1 Definition of the Test Statistic

In this section we describe the multivariate signed-rank test, denoted by Wn, proposed by Peters and Randles (1990) for the one-sample multivariate location problem. As explained in section 2.1, this test can also be applied to repeated measures designs for detecting treatment effects.
Using the same notations as in chapter 2, we let Y, ..-, Yn be i.i.d. from a p-variate elliptically symmetric distribution and Z1, ..., Zn be the transformed sample defined in (2.1.3). Recalling the result of Theorem 2.2.5, we have that Z1, ..., Zn are i.i.d. from a (p-1)-variate elliptically symmetric distribution. Thus, it is logical to measure the distance of each observation Zi, i = 1, ..., n, from the origin in terms of elliptical contours and to use the ranks of these distances along with the observations' directions in forming a test statistic.
We now describe such a signed-rank statistic based on Vn which includes the univariate signed-rank statistic as a special case. Specifically, let us form estimated Mahalanobis distances via


Di = zwe'zi = n, (3.1.1)


where









in T
1 n



is a consistent estimator of the null hypothesis variance-covariance matrix of Z, provided it exists, with H0 defined in (2.1.4). Let Qi = Rank (f)) among D1, ... , i = 1, ..., n. We now weight the (i, k)th term in the sum Vn, defined in (2.1.5), by QiQk, and consider the statistic

n(p-l) n n QiQk
2n=3('l) , cos(t, lk) n n
W n2 ,~~~ff (3.1.2) n i= 1k=l


where Pik is defined in (2.1.6). We reject H0 in favor of Ha for large values of the statistic Wn�

Since the Pik' i, k = 1, ..., n, are invariant with respect to a nonsingular linear transformation (as shown by Randles [1989]) as are the Di, i = 1, ..., n, it is clear that Wn is likewise affine-invariant. When p = 2, the test based on Wn is the two-sided univariate Wilcoxon signed-rank test. For p > 2, Wn does not have a small-sample distribution-free property, but its large-sample null distribution is convenient, as is shown in the next section.

3.2 Asymptotics

In this section we develop the asymptotic distributional properties of nWn under the class of elliptically symmetric distributions. Assume that Y is elliptically symmetric, that is, the density function of Y is of the form

fy(y) = KpIZ1/2h[(y - 1)T-.I'(Y - a)], Y E RP, (3.2.1) where ". is the point of symmetry, I is the variance-covariance matrix of Y, provided it exists, and Kp > 0. The null distribution of nWn is established in the following lemma.











Theorem 3.2.2 Assume the observations Y1, ..., Yn are i.i.d. from a p-variate elliptically symmetric distribution with a density function defined in (3.2.1). Then, in the repeated
2
measurement settings, the test nWn, defined in (3.1.2), has a limiting X , distribution under H0, defined in (2.1.4).

Proof of Theorem 3.2.2 Note that, under H0, taking A YAT =JIp.l, the (p-1)x(p-1) identity matrix, Z1, ..., Zn are i.i.d. as Z = (Zl, ..., Zp.1), where Z is from an elliptically symmetric distribution and can be expressed as Z = RU, where R2, as before, equals Tz and U is distributed uniformly on the (p-1)-dimensional unit-sphere independent of R. (See, e.g., Johnson [1987].) It can be verified that E[(ZjZk)2] = E[R4UjU2 = E[Rn]E[U2U2] < E[R4] < oc, for all j, k = 1, ..., p-1. Thus, via the Lindeberg-Levy Central Limit Theorem (see, e.g., Serfling [1980], p. 28), each element of ''Fn(j - Ip-1),
1 n T
where Z = - Z Zi Zi, is asymptotic normal under H0. Therefore, we have N]n(, - 0
n1=1

= Op(l) under H0. (See, e.g., Serfling [1980], p. 8.) So, the test nWn has a limiting X-2 distribution under H0. (See Peters and Randles [1990], p. 553.)


Next, we derive the Pitman asymptotic relative efficiency of nWn relative to Hotelling's T2. To do this, we first establish the asymptotic distribution of nWn under contiguous alternatives, as described in section 2.3, for a general class of elliptically symmetric distributions.


Lemma 3.2.3 Suppose X1, ..., Xn are i.i.d. from an elliptically symmetric distribution with density of the form


fx (= Kmh[(x - ,)T( L)], x E Rm,












= f I fx U d


for all d r Rm- (0}.


Under the sequence of contiguous alternatives for which X has density of the form fx--d~d'n), we have


d2
nWn d 'Xm


12 f Er _ h_(R2) m H0L h(R2)


H0 implies 1, = , R2 = XTX , h'(R2) = dh(R2)/dR2, and K(t) = PH0(R2 < t), t > 0. Proof of Lemma 3.2.3 Let


[dT.f(Xi)


I n


-n


_ n l T


Sn = ,y


where d, _,E R'-{o, and ijj = [Uil, ..., Uim]T is distributed uniformly on the m-dimensional unit-sphere independent of Ri. Here Xi = Ri~li and Ri = xTx, and we


can write


satisfying


(3.2.4)


where


(3.2.5)


3m F[ i I UiK(R2),
N/n i__Il


Y, UimK(R2) I n i=l1









Tn



4n i=i


dTKmh'(x2j Xi).2Xi

Kmh(_X.Txi) 2h'(R2) RidTU.

h(R2)


Thus, under H0,


= ni


h dm K(R2,
2h'(R 2).-R idrLi /h(R 2)


1 in [Vii (say), Fni L2 [WJ


Vi = -34i K(R2)2Tui and Wi = 2h'(R 2).RidUi /h(R2). Note that, under H, [ V]'s are i.i.d. with


E[ W =0 and Var Wi=
W. I Ly012 02


where


O11 = E(Vi) = 3mE[K2(R2)].E(kjTuiT) = T,


since


E[K2(R2)] = 1 and E(1iJ4) =
3 n (li~T m Im


h'(R ).-Ri
022 = E(Wi)=4E ] 12dT,


ISn [TnJ


where










and

K(Ri)Rih'(R2
Y12 = E(ViWi)=2m hE 2) 4in h(Ri)


Thus,
[S~n dN2 0 1 (Tll 12]udr0 012o.2
[Tj -4N2 ( 0 F12 0221 )under HO.


(See, e.g., Serfling [1980], Theorem B, p. 28.) Applying LeCam's Theorems on contiguity (see Hijek and Sidk [1967], p. 212-213), under the condition defined in (3.2.4), we have

Sn d N(012' o.11)


under the sequence of alternatives. Therefore, under the sequence of alternatives,

dNm( 2" E%[ K(R2)Rh'(R2) ]d, Im), m\ "-4N m EH[KRh(R 2)


and hence
dK(R 2)Rh'(R 2) }2T) nWn H h(R 2)



(See Peters and Randles [1990], Result 1 and Theorem 2, on p. 555 and p. 556, respectively.)


Theorem 3.2.6 Assume Y1 ... Yn are i.i.d. from a density given by (2.3.1). In the

repeated measurement settings, taking A. AT = Ip-l, if 4v + p > 3, then the Pitman asymptotic relative efficiency of nWn relative to Hotelling's T2 is









16v C2
ARE(nWn, T2) = 12 --( _co


f f ffsP2rP-' (r2+t2)v4 expt-[(s2+u2)/Co]V exp{-[(r2+t2)/co]V)ds du dt dr}2
0000

(3.2.7)

where Cp is defined in (2.4.8) and CO, Kp are as defined in (2.3.2). Proof of Theorem 3.2.6 Note that, under H0, taking A_ AT = Ip.1, ZI, ..., Zn are i.i.d. with density of the form
P00 _jzs)C]-1
fZ = Kpg(-)= Kp j exp{-[(zTz+sE)/c0]V}ds, z e R . (3.2.8)
-00


(See Lemma 2.3.3.) Under the sequence of alternatives, described in section 2.3, Z, ..., Zn are i.i.d. with density fz(z - c/',n ), where c e RPI-{0). We have shown that 0 < I;(fz) < oo if 4v + p > 3 (see Appendix A), thus, under the sequence of alternatives, it follows from Lemma 3.2.3 that

nW, d> X2 ( 12 {EH[ K(R2)Rg'(R2) 2T)
nPn Pp.1HO g(R2)



where

g'(R2) = dg(R2)/dR2 and R2 = ZTZ.


Therefore,
K(RE)Rg'(R2) 1 329 ARE(nWn, T2) = 1 f EH[ g(R2) ] 2 (3.2.9)


Recall Lemma 2.4.2 that R2 = zTZ has density of the form










fR2(r) = K-- )/2] r(P1')/2"1g(r), r > 0. (3.2.10) f[(p-l )/21 Thus, we have



E [K(R2)Rg/(R2)]
HO g(R2)



= Lr K r-F '(r)] fR2(r) dr
0

C 0 [ K(r)-fr g'(r)] r(p1)/2.1g(r) dr
0 f g(r)
00

Cp f K(r)g'(r)rP/21 dr.
0

Note that


g'(r) = ( v) f (r+t2)v- expf-[(r+t2)/CO]v} dt.
0


It follows that

EH [K(R2)Rg'(R2)] g(R2)


= ) f fK(r).rP/2"l(r+t2)v-1 exp{-[(r+t2)/Co]v} dt dr (-2v. 0) f . [ f fR2(s) ds ] rP/2-1(r+t2)vl exp{-[(r+t2)/Co]v} dt dr.

Co 00 0

Using the expressions (3.2.10) and (3.2.8) for fR2 and g, respectively, in the above


expression yields











EH0[ K(R2)Rg'(R2)] g(R2)


-4v C2
' ) 7 7s(P1)/2-1 rP/2-1 (r+t2)v'l exp{'[(s+u2)/c0]V}"
0O 0000

exp{-[(r+t2)/Co]V}ds du dt dr

-16v C 2
-P P) j j fsp'2rp'I (r2+t2)v'1 exp{'[(s2+u2)/C0]V}"

C0 0000


exp{-[(r2+t2)/C0]v} ds du dt dr.

Substituting the above expression into (3.2.9) yields Theorem 3.2.6.




3.3 Numerical Evaluation of ARE(nWn, T2)


In this section we describe the numerical evaluation of the ARE expression given in Theorem 3.2.6. All calculations were performed with an IBM computer running on a VM/CMS operating system using fortran 77. Simpson's rule and the IMSL subroutine DMLIN were used to integrate single integrals and 3-dimensional integrals, respectively.

Now we describe how the calculations were performed as well as the error associated with them. Letting
* t* r*
u=eu, et, andr* =er, we can write

-ARE(nWn, T2)








S1 62 CP. ) "J "J P2(-/rnr*)p'I (r*t*u*l'l[(!'nt*12+(!'nr*)2]v'l"
1 C2 *

COV 000 0
exp f-[(s2+(!nu*)2)/Co]v } exp {-[(('nr*)2+(nt*)2)/Co]V) ds du*dt *dr* }
-" [ I j4CpsP-2ulexp{-[(s2+(inu)2)/Cov} ds
000 0

l4v (fnr)pl[(nt)2+(nr)2]vl(tr)lexp{[((In)2+([)2)/0IVldu dt dr
/1[- Inr
f f k(u, s) ds] h(t, r) du dt dr, (3.3.1)


where

k(u, s) = 4CpsP'2u" exp {[(s2+(nu)2)/C]vI ,


h(t, r) = 4PC (-rnr)Pl[(fg)2+(fr)2]vl(t r)lexp{[(([)2+(rnt)2)/C]V}
0

and In is the natural logarithm. Note that, following standard arguments and using expressions (2.4.7) and (2.4.9),
1 1
f f h(t, r) dt dr = /3ARE(Vn, T2), (3.3.2) 00

where ARE(Vn, T2) is given in Theorem 2.4.5. Using Simpson's rule with an absolute error cj/ 3ARE(Vn, T2), we can approximate the inner single integral in expression (3.3.1) by g(u, r) satisfying

-f k(u, s) ds - g(u, r) I < E1/ 3ARE(Vn, T2). (3.3.3)
0









Hence, using expressions (3.3.2) and (3.3.3), we can write expression (3.3.1) as

ARE(nWn, T2)

f111 [g(u, r) � el/" 3ARE(Vn, T2) ] h(t, r) du dt dr
000
111 rI 11
f f f g(u, r)h(t, r) du dt dr � [e/4 3ARE(Vn, T2) ] f f h(t, r) dt dr
000 00

111
f f ff g(u, r)h(t, r) du dt dr � el. (3.3.4)
000

Now, using the IMSL subroutine DMLIN with an absolute error C2, we can approximate the above 3-dimensional integral by VAL satisfying f f 1 g(u, r).h(t, r) du dt dr - VAL I < e2. (3.3.5) 000

Combining expressions (3.3.4) and (3.3.5), we have ARE(nWn, T2) = VAL � (P1+-2), where VAL satisfies expression (3.3.5). Thus, we approximate ARE(nWn, T2) by (VAL)2 with the maximum error

ERREST = 2VAL(e1+E2) + (el+62)2. (3.3.6) In Table 3.1 we present the asymptotic relative efficiencies for nWn with respect to T2 for selected values of v and p satisfying the condition 4v+p > 3. We also include ARE(Vn, T2), in parentheses, for easy comparison. When the underlying population is multivariate normal (v = 1.0), Hotelling's T2 performs well. Both the sign test and signed-rank test appear to be quite competitive, with the signed-rank test slightly better than the sign test provided the dimension is not too large. For light-tailed distributions









(v = 2.0 and 5.0) and p > 2, the signed-rank test Wn has greater power compared to both Hotelling's T2 and Vn. For heavy-tailed distributions (v = .50 and. 10), Vn is clearly most powerful, although, Wn still performs well relative to Hotelling's T2, provided the dimension is not too large.
In Table 3.2 we display the values of ERREST, defined in expression (3.3.6), which bound the error in the estimates of ARE(nWn, T2). Assuming c1 = C2 = e, where el and e2 are defined in (3.3.3) and (3.3.5), respectively, we use the foiling e values : e = .001 when v = 1.0, e = .001 when v = 2.0 and p =2 to 6, c = .005 when v = 2.0 and p=7or8, e= .01 whenv =5.0andp =2 to6, e= .02 when v =5.0andp =7or8, E = .001 when v = .50, e = .10 when v = .10 and p = 2, and e = .015 when v = .10 and p=3 to 8.









Table 3.1
ARE(nWn, T2), with ARE(Vn, T2) in parentheses.


V
p 5.0 2.0 1.0 .50 .10


2 .8674 (.4205) .8779 (.4784) .9550 (.6366) 1.2658 (1.2159) 36.4044* (297.1379*) 3 1.0834 (.5738) 1.0124 (.6364) .9855 (.7854) 1.0786 (1.2337) 8.4010 ( 50.7101) 4 1.1809 (.6599) 1.0535 (.7191) .9748 (.8488) .9873 (1.2008) 4.1877 ( 20.1730) 5 1.2208 (.7164) 1.0664 (.7708) .9614 (.8836) .9347 (1.1711) 2.7726 ( 11.4238) 6 1.2476 (.7566) 1.0683 (.8063) .9491 (.9054) .9012 (1.1477) 2.0814 ( 7.7606) 7 1.2525 (.7868) 1.0593 (.8323) .9382 (.9204) .8777 (1.1295) 1.7042 ( 5.8638) 8 1.2574 (.8103) 1.0592 (.8521) .9289 (.9313) .8605 (1.1151) 1.4895 ( 4.7408)

*Conjectured ARE's when p = 2 and v < .25, using the expressions in Theorem 3.2.6 and Theorem 2.4.5
for ARE(nWn, T2) and ARE(Vn, T2), respectively.









Table 3.2
Error Estimate ERREST of ARE(nWn, T2)



v
p 5.0 2.0 1.0 .50 .10


2 .0377 .0038 .0039 .0045 2.4534 3 .0420 .0040 .0040 .0042 .1748 4 .0439 .0041 .0040 .0040 .1237 5 .0446 .0041 .0039 .0039 .1008 6 .0451 .0041 .0039 .0038 .0875 7 .0911 .0207 .0039 .0038 .0792 8 .0913 .0207 .0039 .0037 .0741














CHAPTER 4
MONTE CARLO STUDY


In this chapter we display results from a Monte Carol study when the dimension is p = 4, the sample size is n = 20, and the significance level is a, = .05. In addition to the affine-invariant statistics T2, Vn, and nWn, we examined two other nonaffine-invariant statistics. The ANOVA F test is included along with the rank transformation test RT, which were both introduced in chapter 1. These five test statistics were compared under five different distributions. They were quadrivariate normal distribution, elliptically symmetric distributions with density of the form given in (2.3.1), Pearson Type II and Type VII (see Johnson [1987], p.111 and p.117-118, respectively), and the mixtures of quadrivariate normal distributions. (A quadrivariate normal mixture is obtained by selecting randomly one of two quadrivariate normal distributions. Each of the observations is sampled with probability p from the first distribution and with probability 1-p from the second distribution. See Johnson [19871, p.56-57.) These distributions were located at lia = (m5, m8, mS, 0), m = 0, 1, 2, 3, for the original sample Y1I, ..., Yn, on which ANOVA F test and rank transformation test RT were applied, and they were located at % = (m, mS, mS), m = 0, 1, 2, 3, for the transformed sample Z1, ..., Zn, on which the tests t, Vn, and nWn were performed. The value of 8 was adjusted for different distributions to examine somewhat silimar points on the power curves. Since the performances of tests ANOVA F and RT depend on the variance-covariance structure of the distribution, for each of the above first four distributions, we considered two types of variance-covariance matrices, one with I4, the identity matrix, and the other one with a non H-type structure. For the mixtures of normal distributions, we consider one mixture with a non H-type structure, and three other mixtures with H-type structures. In each Monte Carlo simulation,









the proportion of times out of 1000 in which each test statistic exceeded the upper a-percentile of its null distribution is reported. The asymptotic null distribution X2 is used to determine the critical value for tests Vn and nWn. For tests ANOVA F and rank transformation RT, the null distribution F3,57 is used to determine the critical value. While 57
for Hotelling's T2, we use the null distribution -jjF3,17 (namely, a multiple of F3,17) to determine the critical value. All calculations and random variables generations were performed with an IBM computer running on a VM/CMS operating system using fortran 77. Several IMSL subroutines, to be described later in this chapter, were used.

In Tables 4.1 and 4.2, the results from the Monte Carol studies for quadrivariate normal distributions with Y =I4 and I = E ET, respectively, are presented. Note that we used

1 3 0 2
E= - 2 1 4 -1
1 -1 1 0'(41)
1-2 4 -1 3

producing
14 -1 -2 16ET -1 22 1 1
EE = 2 1 3 -7 (4.1.2) 16 1 -7 30


which is non H-type. The IMSL subroutine GGNML was used to generate N(0, 1) variables. As indicated, 1000 samples of size 20 and significance level .05 were used. When Z = 14, as we may expect, ANOVA F and rank transformation test RT have better power than Hotelling's T2. And they all perform better than Vn and nWn (see Table 4.1). However, when Z - ET, Hotelling's T2 has good power, followed by Vn and nWn, both performing better than F and RT (see Table 4.2). This illustrates the strong dependence of the performances of tests F and RT on the variance-covariance structure of the distribution. For both variance-covariance structures, ANOVA F performs better than RT, and nWn









performs slightly better than Vn, which agrees with the result found in Table 3.1 for v =1.0 and p = 4.
In Tables 4.3 and 4.4, we display the Monte Carlo results for five members of the class of elliptically symmetric distributions with density of the form given in (2.3.1) with = 14 and Y = E ET, respectively. Heavy-tailed distributions, v = .10 and .50, the quadrivariate normal distribution, v = 1.0, (repeated in this table to permit comparisons), and light-tailed distributions, v = 50.0 and 100.0, were included. Taking L = 0 and L = 14 in (2.3.1), it follows from standard arguments that P(R2 < w) = P(C0G1/v < w), w > 0,

where R2 = yTy, CO is defined in (2.3.2), and G has the distribution of Gamma(l, p/2v). Thus Y can be generated via Y = R U, where R is independent of U having the distribution of f/-oG1/2v, and U is uniformly distributed on the unit-sphere Sp (see Johnson [1987], p.110). Here, the IMSL subroutines GGAMR and GGSPH were used to generate Gamma(l, r) and the variables that are uniformly distributed on the unit-sphere Sp, respectively. Although it is difficult to compare powers for statistics with different rejection proportions under the null hypothesis, it appears that the Monte Carol results tend to agree with the asymptotic results of the previous two chapters. For light-tailed distributions (v = 50.0 and 100.0), ANOVA F and RT have great power ifI = 14, while Hotelling's T2 and nWn have the greatest power when Y, = Ee. For the heaviest-tailed distribution (v = 10) examined, Vn performs best, followed in order by RT and Hotelling's T2 when Y, = 14, and by Hotelling's T2 and RT when Y, = E ET. Generally speaking, for light-tailed distributions (v = 50.0 and 100.0), the signed-rank test nWn performs better than the sign test Vn, and ANOVA F performs better than RT; however, for heavy-tailed distributions (v =. 10 and .50), the interdirection sign test Vn performs better than nWn, and RT works better than F. The superiority of tests ANOVA F and nWn (RT and Vn) for very light-tailed (heavy-tailed) distributions is shown here. Except for the heaviest-tailed distribution (v = .10), ANOVA F









and RT both perform better than Vn and nWn when I = 14, however, Vn and nWn both perform better than ANOVA F and RT when I = E ET. Also, we notice that RT performs better than Hotelling's T2 only when I = 14.

Next, we consider the light-tailed Pearson Type II distributions, a special case of elliptically symmetric distributions, with the density function = F(m+3) l 1 )T.I1.)}m (4.1.3) fY(Y) = I m)l In,


having support (y-aT_-l(.) < 1 and shape parameter m > -1. The Monte Carlo results for this family of distributions with 1 =14 and Y, = E ET are presented in Tables 4.5 and 4.6, respectively. Taking Uj = 0 and Z = 14 in (4.1.3), it can be shown that R2 = yT Y has the distribution of Beta(2, m+l). Thus, the variable generation is similar to that in the last case considered. (See, Johnson [1987], p. 116.) Here, the IMSL subroutine GGBTR was used to generate beta variables. An examination of Tables 4.5 and 4.6 indicates that ANOVA F and RT have great power if I = 14, while Hotelling's T2 and nWn have the greatest power when I = E ET. The signed-rank test nWn performs better than the sign test Vn, and ANOVA F performs better than RT. Furthermore, ANOVA F and RT both perform better than Vn and nWn when Y, = 14, while Vn and nWn both perform better than ANOVA F and RT when I = E ET. These results agree with the findings in Tables 4.3 and

4.4 for the light-tailed elliptically symmetric distributions.

For the heavy-tailed Pearson Type VII distributions, the Monte Carlo results are presented in Tables 4.7 and 4.8 for I = 14 and I = EET, respectively. The density function has the form of

f(y) m) '/2{ + (Y-)Tj'I -M (4.1.4)



where the shape parameter m > 2. Note that, taking 1 = 0 and 1 = 4 in (4.1.4), Y can be generated via Y = Z/F (see Johnson [1987], p. 118), where Z is quadrivariate normal









with mean 0 and variance-covariance matrix 14, and S is independent of Z having the distribution of X2, b = 2m - 4. Note that when m = 2.5, Y has a multivariate cauchy distribution. In Tables 4.7 and 4.8, the sign test Vn has the greatest power. Notice that Vn performs better than nWn, and RT works better than ANOVA F. Also, Vn and nWn both perform better than F and RT when Y = E ET. Note that the same general trends as in the heavy-tailed elliptically symmetric distributions are seen here.

Finally in Table 4.9 we examine samples from quadrivariate normal mixtures violating the assumption of elliptical symmetry required for the asymptotic results of the previous two chapters. As we may expect, the sign test Vn and signed-rank test nWn appear to be quite robust. For each mixture in Table 4.9, the parameters of the distributions (ul, A2, Z1, and Y,2) and the mixing probability (p) as well as the parameters of the mixture (U, and Z), computed by

L = PI + G1-P)U2,

and


p =Pl1 +(lP)y+ P(-P)(1l -UI2)(Ul1 _ L2)T, (4.1.5) (see, Johnson [1987], p.57), are indicated. In mixture 1, a light-tailed distribution with H-type structure, we see that tests ANOVA F, RT, and Hotelling's T2 have great power, and all performing better than Vn and nWn. This agrees with that of Table 4.1, for normal distribution with I = 14. An examination of the table for mixtures 2 and 3, heavy-tailed distributions with H-type structure, shows the same general trends. That is, the superiority of tests Vn and RT over the signed-rank test nWn and the other competitors are seen. Also, by comparing with the result for elliptically symmetric distribution with v =. 10 and = in Table 4.3, and that for Pearson Type VII distribution with m = 3.0 and I = I4 in Table 4.7, it illustrates that the performances of Hotelling's T2 and ANOVA F are affected by the









asymmetry of the distributions, which as a result makes nWn a better test than Hotelling's T2 and ANOVA F. Finally, in mixture 4, a heavy-tailed distribution with non H-type structure, the robustness and superiority of tests Vn and nWn is easily seen.

In summary, for normal to light-tailed symmetric distributions and a variancecovariance structure that is non H-type, Hotelling's T2 has the greatest power, followed by the signed-rank test nWn. When the structure is H-type and the populations are normal or light-tailed the best is ANOVA F, followed by RT. For the heavy-tailed distributions with H-type variance-covariance structure, the interdirection sign test Vn has the greatest power, followed by RT. For heavy-tailed symmetric distributions with non H-type variancecovariance structure, the interdirection sign test Vn still has the greatest power, but this time Hotelling's T2 is second best. For the asymmetric distribution with non H-type variancecovariance structure, the interdirection sign test Vn and the signed-rank test nWn are the best. Generally, the tests Vn and nWn both perform better than ANOVA F and RT when the variance-covariance structure is non H-type while the contrary is true only for normal to light-tailed distributions with H-type variance-covariance structures. This suggests that Vn and nWn are promising procedures for the repeated measures settings. The tests ANOVA F and RT follow the same general patterns as nWn and Vn, respectively. That is, the RT (Vn) test performs better than ANOVA F (nWn) for heavy-tailed distributions, and ANOVA F (nWn) performs better than RT (Vn) for light-tailed distributions. Also note that test RT works better than Hotelling's T2 for distributions with H-type variance-covariance structures. Finally, we should point out that the performance of the test based on nWn is disappointing, since the power of nWn is not always better than that of Hotelling's T2 for light-tailed symmetric distributions (see Tables 4.3 and 4.4). One explaination to this may be because of the higher probability of Type I error of Hotelling's T2 test.











Table 4.1
Monte Carlo Results for Quadrivariate Normal Distribution with I = 14.


Statistics
Amount
of F RT T2 Vn nWn
Shift

.00 .049 .046 .056 .052 .058 .25 .106 .113 .108 .082 .108 .50 .335 .321 .300 .219 .247 .75 .649 .644 .579 .457 .485


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution. Dimension: p = 4. Sample Size: n = 20. Number of Samples: rep = 1000. Significance Level: a = 0.5.











Table 4.2

Monte Carlo Results for Quadrivariate Normal Distribution with , = E ET.


Statistics
Amount
of F RT T2 Vn nWn
Shift

.00 .081 .092 .055 .040 .056 .60 .091 .101 .108 .088 .114 1.20 .164 .144 .283 .204 .235 1.80 .279 .248 .540 .430 .453


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution. Dimension: p = 4. Sample Size: n = 20. Number of Samples: rep = 1000. Significance Level : ac = 0.5.











Table 4.3

Monte Carlo Results for Elliptically Symmetric Distributions with Density of the Form Given in (2.3.1) with I = 1.


Statistics
Amount
of F RT a Vn nWn Shift

v = .10

.00 .026 .048 .016 .032 .040 .17 .079 .226 .097 .276 .124 .34 .248 .636 .367 .669 .288 .51 .500 .872 .626 .894 .432

v =.50

.00 .053 .057 .050 .032 .048 .25 .110 .119 .101 .079 .088 .50 .321 .368 .303 .260 .234 .75 .684 .699 .618 .573 .462 v = 1.0

.00 .049 .046 .056 .052 .058 .25 .106 .113 .108 .082 .108 .50 .335 .321 .300 .219 .247 .75 .649 .644 .579 .457 .485










Table 4.3 - - continued.


Entries : Proportion of times each test

null distribution. Dimension: p = 4. Sample Size: n = 20. Number of Samples: rep = 1000. Significance Level : a = 0.5.


statistic exceeded the upper a-percentile of its


Statistics
Amount
of F RT a Vn nWn Shift

v = 50.0

.00 .046 .049 .059 .032 .044 .25 .100 .085 .092 .057 .087 .50 .316 .279 .247 .166 .261 .75 .656 .594 .559 .361 .568 V = 100.0

.00 .050 .053 .053 .032 .048 .25 .095 .089 .091 .056 .088 .50 .317 .282 .256 .159 .262 .75 .650 .584 .573 .354 .557











Table 4.4
Monte Carlo Results for Elliptically Symmetric Distributions with Density of the Form Given in (2.3.1) with = EET.


Statistics
Amount
of F RT T2 Vn nWn Shift

v = .10

.00 .055 .081 .021 .049 .037 .40 .073 -.128 .094 .277 .127 .80 .124 .277 .325 .657 .268 1.20 .224 .462 .586 .879 .418 v =.50

.00 .079 .081 .060 .049 .063 .60 .094 .104 .110 .100 .107 1.20 .154 .152 .300 .266 .238 1.80 .268 .219 .609 .558 .452 v = 1.0

.00 .081 .092 .055 .040 .056 .60 .091 .101 .108 .088 .114 1.20 .164 .144 .283 .204 .235 1.80 .279 .248 .540 .430 .453









Table 4.4 - - continued.


Statistics
Amount
of F RT Vn nWn
Shift

v = 50.0

.00 .078 .079 .060 .049 .048 .80 .110 .105 .139 .098 .124 1.60 .197 .163 .438 .270 .434 2.40 .390 .275 .833 .620 .830

v = 100.0

.00 .073 .080 .065 .049 .055 .80 .106 .109 .138 .099 .123 1.60 .203 .157 .428 .275 .447

2.40 .394 .280 .824 .617 .822


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution.

Dimension: p = 4.

Sample Size: n = 20.
Number of Samples : rep = 1000. Significance Level: a = 0.5.











Table 4.5

Monte Carlo Results for Pearson Type II Distribution with Shape Parameter m, and Density of the Form Given in (4.1.3) with 2; = 14.


Statistics
Amount
of F RT T2 VnnW
Shift

m= 1.0

.00 .045 .045 .050 .032 .039 .10 .115 .106 .108 .064 .092 .20 .402 .345 .340 .221 .292 .30 .771 .724 .687 .493 .649 m = 2.0

.00 .052 .045 .051 .032 .053 .10 .136 .130 .122 .077 .113 .20 .498 .451 .415 .281 .370 .30 .867 .830 .788 .601 .726


Entries Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution.

Dimension: p = 4.

Sample Size: n = 20.

Number of Samples : rep = 1000. Significance Level: a = 0.5.











Table 4.6

Monte Carlo Results for Pearson Type II Distribution with Shape Parameter m, and Density of the Form Given in (4.1.3) with = E ET.


Statistics
Amount
of F RT T2Vn nWn
Shift

_ m= 1.0

.00 .078 .080 .060 .049 .053 .30 .111 .109 .153 .111 .144 .60 .223 .170 .490 .325 .454 .90 .438 .330 .877 .704 .832 m = 2.0

.00 .072 .075 .059 .049 .054 .30 .118 .113 .172 .125 .160 .60 .275 .195 .587 .425 .522 .90 .540 .403 .930 .812 .885


Entries : Proportion of times each test statistic exceeded the upper oa-percentile of its

null distribution.

Dimension : p = 4.

Sample Size: n = 20.

Number of Samples: rep = 1000. Significance Level: ot = 0.5.











Table 4.7

Monte Carlo Results for Pearson Type VII Distribution with Shape Parameter m, and Density of the Form Given in (4.1.4) with 1_ = 14.


Statistics
Amount
of F RT a Vn nWn
Shift

m =_2.5

.00 .019 .052 .022 .052 .050 .40 .024 .106 .042 .107 .097 . 80 .067 .262 .109 .287 .190 1.20 .120 .476 .214 .520 .312 m = 3.0

.00 .035 .046 .030 .052 .058 .50 .179 .324 .226 .295 .220 1.00 .592 .863 .661 .829 .575 1.50 .851 .985 .901 .976 .777


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution.

Dimension: p = 4.

Sample Size: n = 20.
Number of Samples: rep = 1000. Significance Level: a = 0.5.











Table 4.8

Monte Carlo Results for Pearson Type VII Distribution with Shape Parameter m, and Density of the Form Given in (4.1.4) with Z = E ET.


Statistics

Amount
of F RT T2 Vn nWn
Shift

m =_2.5

.00 .025 .077 .027 .040 .048 1.20 .031 .098 .058 .135 .110 2.40 .054 .173 .149 .376 .232 3.60 .087 .294 .281 .653 .375 m_ = 3.0

.00 .059 .080 .027 .040 .051 1.10 .093 .137 .182 .251 .199 2.20 .228 .340 .567 .709 .508 3.30 .442 .620 .848 .939 .715


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution.
Dimension: p = 4.

Sample Size: n = 20.
Number of Samples : rep = 1000. Significance Level : a = 0.5.











Table 4.9
Monte Carlo Results for Quadrivariate Normal Mixtures with Mean JL and VarianceCovariance Matrix I given in (4.1.5).


Statistics
Amount
of F RT T2 Vn nWn Shift


Mixture 1
= .5, 2 I ,1 1T=-d2 g 14, 2111].
1211
Id'0' =1 1 2 1



.00 .049 .051 .047 .033 .049 .35 .190 .179 .171 .121 .160 .70 .604 .565 .528 .405 .428 1.05 .929 .915 .896 .800 .770


Mixture 2
p=.9, Ul=U2=O, -I =14, -Z2=400._4, = E_ = 40.9.14.

.00 .021 .050 .031 .046 .050 .35 .037 .151 .078 .102 .125 .70 .096 .427 .231 .336 .317 1.05 .166 .744 .423 .668 .589









Table 4.9 - - continued.

Statistics
Amount ,,,
of F RT T2 VnnW Shift
Mixture 3
p = .5, UI [,1 1, 1]T = _U2, �--1 =1I4, Z2 =400"14, 201.5 1 1 1 201.5 1 1
[201 1 201.5 1
1 1 1 201.5


.00 .035 .050 .024 .033 .038 1.40 .055 .229 .047 .332 .149 2.80 .090 .444 .088 .654 .216 4.20 .172 .543 .175 .769 .212

Mixture 4
p=.9, II = U2=0, 11 =EET-2= 100"EET, ,=0 = 10.9"E ET EET is defined in (4.1.2).


.00 .033 .072 .030 .035 .053 1.60 .076 .144 .219 .292 .295 3.20 .224 .402 .617 .834 .723 4.80 .395 .738 .808 .986 .880


Entries : Proportion of times each test statistic exceeded the upper a-percentile of its

null distribution.

Dimension : p = 4.

Sample Size: n = 20.
Number of Samples : rep = 1000.






65


Table 4.9 - - continued. Significance Level: a = 0.5. Distributions : Quadrivarite normal mixture, choosing N1 with probability p and N2 with

probability 1-p, where Ni, i =1, 2, is quadrivariate normal with mean ji and variance-covariance matrix L. The resulting mixture has mean 1L and

variance-covariance matrix Y, (see (4.1.5)).














CHAPTER 5
A MULTIVARIATE SIGNED SUM TEST BASED ON INTERDIRECTIONS
FOR THE ONE-SAMPLE LOCATION PROBLEM



5.1 Definition of the Test Statistic

We now leave the repeated measures problem, and consider the one-sample multivariate location problem. In this section we propose a multivariate signed sum test based on interdirections, described in section 2.1, for the one-sample multivariate location problem. To do so, we let X1, ..., X.n, where Xi = (Xil, ..., Xp) T, be i.i.d. as X = (X1,

..., Xp)T, where X is from a p-variate absolutely continuous population with location parameter Q* (pxl). We would like to test H0: Y =Q versus Ha:O*Q. (5.1.1) Here Q is used without loss of generality, since H0 : Q* = 00 can be tested by subtracting 0 from each observation 2, and testing whether the differences (Y- - 9i)'s are located at
0.

The procedure is somewhat like applying the multivariate sign test, proposed by Randles (1989) for the one-sample location problem, to the sums S+ .t, 1
where









- Ciki k+dn if (i, k) (i', k')
- (P1)


0 if (i, k) = (i', k'), (5.1.4)

and Cik,i'' denotes the number of hyperplanes formed by the origin 0 and p- I of the other observations 2j (excluding X i, , Xi', and 4k') such that Xi + Xk and Xi, + . are on opposite sides of the hyperplane formed. The counts [ Cki I 1 < i < k < n, 1 < i' < k' < n), called interdirections, are used via tik,il, to measure the angular distance between X, +K and -Xi' + Xk' relative to the positions of the other observations. Here i 'k' is the observed fraction of times that 2Xi + & and 2i' + ' fall on opposite sides of the hyperplanes formed by Q and other p-I observations.
We now examine some characteristics of the test based on SS. First of all, it can be easily seen that the test SS defined in (5.1.3) is like the multivariate sign test Vn applied to the sums 2s + X-t, 1 < s 5 t < n. Secondly, since the ik,i'k, 1 i < k < n, 1 < i' < k' < n, are invariant with respect to the nonsinglar linear transformations, as shown by Randles (1989), it is clear that SS is also affine-invariant. Thirdly, for p > 1, SS does not have a small sample distribution-free property. This is because the joint distribution of the directions of Xi + &, 1 < i < k 5 n, from a depends on the distribution of the distances of Xi 1 < i < n, from the origin. However, its large sample null distribution is convenient, as is shown in the next section. We end this section by proving a Lemma which shows that for p = 1, the test based on SS is the two-sided univariate Wilcoxon signed-rank test. Hence for p = 1 it does have a small sample distribution-free property, as well as a convenient large sample null distribution.


Lemma 5.1.5 When p = 1, the test based on SS is the two-sided univariate Wilcoxon signed-rank test.










Proof of Lemma 5.1.5 Note that when p = 1,


and thus


1 if (Xi + Xk)(Xi, + Xk') < 0

Pik,i'k' = { if (Xi + Xk)(Xi, + Xk,) = 0
0 if (Xi + Xk)(Xi,+ Xk,) > 0



-1 if (Xi + Xk)(Xi, + Xk,) < 0 Cos(1rlcik') = { 0 if (Xi + Xk)(Xi, + Xk') = 0
1 if (Xi + Xk)(Xi, + Xk') > 0


Hence, we can write

cos(1t~ikik,) = sign (Xi + Xk)'sign (Xi' + Xk') where sign(x) = -1 if x < 0, 0 if x = 0, and 1 if x > 0. Therefore we have


- n4n) 1 cos(Or Pik,ik')
n(n+ i i'

nn 4 Y Ysign (Xi + Xk).sign (Xi, + Xk,) n(n+ ik i'

(5.1.6)


_ 4 { Zsign (X, + Xk)}2.
n(n+)2ik


Noting that the Wilcoxon signed rank test W+ is equal to the number of positive Walsh averages ((Xi + Xk)/2, 1 < i < k < n) (See, e.g., Randles and Wolfe [1979], p.57 and p.83), thus we have


I sign (Xi + Xk) = W+ - W-, i:5k









where W- is equal to the number of negative Walsh averages ((Xi + Xk)/2, 1 < i < k < n).


Now using the fact that with probability one

- n(n+1)
2'

expression (5.1.6) is equivalent to

SS n)2 {W+-W-12
n(n+1 )

4 _[2+_ n__l_n(n+1)2 {2W+2n( +l)}2

16 _W+_ n(n+l)12
n(n+l)2 4

(W+_ n(n+l))2 n(n+l)(2n+l)
4 24 n(n+l)(2n+l) n(n+l)2
24 16


(W+- EH0(W+))2

VarHO(W+)


2(2n+1)
3(n+l)


since


EH(W+) = n(n+l) a n(n+l)(2n+l)
4 VarH0(W+) 24


(See, e.g., Randles and Wolfe [1979], p.56.) This completes the proof.


Note that, when p = 1,

3 d 2
---SS - X , under HO as n -*oo.


This follows from the fact that









W+- EHO(W+) d
-4 N(O, 1) under H0 VarH0(W+)

2(2n+1) 3
(See Randles and Wolfe [1979], p.85), and 3(n+l) - as n - oo.





5.2 Some Intermediate Results

In this section we present some basic and important results which will be useful in the next section for finding the limiting distribution of SS under H0. Here we discuss some properties involving the sum observations Xi + Xk, 1 < i < k < n, when the original sample Xi, 1 < i < n, is from the family of elliptically symmetric distributions. The first one is about the sum and difference of two i.i.d. spherically symmetric random vectors, a special case of elliptically symmetric distributions.


Theorem 5.2., Let X1, X2 be i.i.d. spherically symmetric random vectors. Then XI+X2 and Xl-X2 are spherically symmetric random vectors. Proof of Theorem 5.2.1 Let D be any pxp orthogonal matrix. It suffices to prove that

__d X_2X
~(i 2) = XI� X2

(see, e.g., Muirhead [19821, p. 32), where d is read "has the same distribution as". Note that we have
d d
X1 d D X1 and X2 A D X2, since X1, X2 are spherically symmetric. Hence, by the independence of X1 and X2, we have

d









Therefore, see, e.g., Randles and Wolfe (1979) , p. 16, g(.X1, X2) d gMXI, D X2), where g(, y) = x + y. Hence

d
X1 + X2 = D X, + D X2 = D(_XI+ X2). The difference is handled similarly. This completes the proof.


Next we consider the general case of two i.i.d. elliptically symmetric random vectors.


Theorem 5.2.2 Let X1, X2 be i.i.d. elliptically symmetric random vectors. Then XI+X2 and X1-X2 are elliptically symmetric random vectors. Proof of Theorem 5.2.1 Since X1, X2 are i.i.d. elliptically symmetric random vectors, we can write

Xi =AX1 and X2 = AX2, where X1, X2 are i.i.d. spherically symmetric random vectors and A is a pxp nonsingular matrix. Thus

Xi� X2= A X1 A X2 = A(K� X2). This completes the proof since X1+ X2 are spherically symmetric random vectors, as shown in Theorem 5.2.1.


Define









h(-lI, X2) = +X2 u' (5.2.3) where
"II'=~~~ 11 (~l2l2, I (t1.,t) j=1

We examine the property of h(.L, X2) in the next theorem. To Let X1, �2 be i.i.d. spherically symmetric random vectors. Then h(, X2) is uniformly distributed on the p-dimentional unit sphere Sp. Proof of Theorem 5.2.4 Since X1, X2 are i.i.d. spherically symmetric random vectors, X1+X2 is also spherically symmetric, as shown in Theorem 5.2.1. Thus it follows that xl+x2
h(Xp,12) = X k1+X2 IT ' U(Sp). (See, e.g., Muirhead [1982], p. 38, Theorem 1.5.6.) The proof is complete.


Note that under the assumptions of Theorem 5.2.4

EH0[h(�I, X2)] =.0 = ExI(Ex2[hX1, X2) I X1]). Define

h*i ) = EX[h(Xi, Xk) I Xi], 1 _< i:5 k5

Theorem 5.2.6 Let X1, X2 be i.i.d. spherically symmetric random vectors. Then h*(l) is a spherically symmetric random vector. Proof of Theorem 5.2.6 Let D3 be any pxp orthogonal matrix. It suffices to show that








D h*(X) A ( h*(XI).

(See, e.g., Muirhead [1982], p. 32.) Note that

D h* (X1) = D Ex[h(Xi, X2) I Xi]

= DEx2[ 11'+X2- I X] (by expression (5.2.3)) D X1+D2

EX2L E XI+DX2 IX1] (111D11 = 1 since D is orthogonal) = Ex2[h(LD X1, D X2) I XI] (by expression (5.2.3))

Ex2[h(_D X1, X2) I X1] ( D X2 d X2 by spherical symmetry) = h*(D XI) (by expression (5.2.7))

d h(_Xl) (because D X1 = X1 by spherical symmetry). (5.2.7) This completes the proof.


Hence, for i.i.d. spherically symmetric random vectors X1 and X2, it follows that we can express h*(X1) as h*(X) = R1 U*,

where UI - Uniform(Sp) is independent of R1, a positive random variable. Defining T2= EHo[(h* (X))Th*(Xl)], (5.2.8)


we get








Var-CovH0[h ()] "- EH0[(R1 U1)(R1 U1)T] (since EH0(h*(X1)) =Q) SEHo[(R1 )2]E[U*(U*)T]


- EHO[(Ri )2].jIr2



since
,r HO(* U*)T (* ,2 = EH0[(R1 U1)T(R1U1)] = EH0[(R1 )2]. In summary we have the following result : if X1, ..., Xn are i.i.d. spherically symmetric random vectors, then h*(X1), ..., h*(Xn), defined in (5.2.5), are i.i.d. spherically symmetric random vectors with
EH[h*a1)] = 0 and Var-CovHjh*(X1)] -where T2 is defined in (5.2.8). Expressing Xi =RiUi and h*(Xi) = R, U., 1


Theorem 5.2.9 Let X1, X2 be i.i.d. spherically symmetric random vectors with Xi = RiUi, i = 1, 2, where Ui." Uniform(Sp) is independent of Ri. Then we can write


*(-X1) = R*(R1)Ul, (5.2.10)


where







R*(RI) = ER2, U21 t


2 2
(R1/R2+U21) +1-U21


is independent of U.1, and U2 = (U21, ..., U2p)T. Proof of Theorem 5.2.9 Let X = R U and h*(X = R* U*. First we want to show that U* = 1. Define


UO= (1, 0, ..., O)T and 0 = RUO.


Then


X__+X2
h* (-g�) = EX [11 o2C 0 I-o] (by expression (5.2.5))


A]


= ER2, U2{


= ER2, U2 {


(R+R2U21)2+R U22+..+


A]


22
(R/R2+U21) +U22..


"R/R2+U21U22

PU -


I


/-C(/( 2+U'21) 2+ l-U 21


(5.2.11)


= ER2, U2 t








P 2)
(since Y U2j= 1) j=1 2

-R/R2+U21U22
= ER2, U21 f EU22, ... I U2p / (R/R2+U21)2+l-U21 L U2p IR2, U21] I1. Noting that, given R2 and U21, the expected value of U2j,j = 2, ..., p, is zero and thus we write the above expression as


= ER2,U21 {


(R/R+Ua? +1-U21


= ER2,U21 I


R/R2+U21
4(R/R2+U21)2 l-21


= R*(R) U,


1--i


0L:


(5.2.12)


where R* is as defined in (5.2.11). Let D be any pxp orthogonal matrix. Now we need to show that


h*(D -X0) = R*(R).D .


Note that


= E X2[h( Xo, X2) I Xo]


= E D X0+X2 I X0
E X2[ 11 D X..0+ X2,1








D Xg+D X2
=Ex2[' D 1X0+D X2 I X] (since X2 diDX2)

= , -' _. 2 I XO] (1I 11 = 1 since D is orthogonal) X2111- Xo+X2 11 0
XXo+X2 I
= D2 %211 X+2L 1


= D h*(XO) = D R*(R) LU (follows from (5.2.12))

= R *(R)'D .Uo. (5.2.13)

Using expressions (5.2.12) and (5.2.13), we have shown that whenever X = R VU, h*(_X) = R*(R) U. The indenpendence of R*(R) and U follows from the independence of R and U and the fact that R *(R) depends on R only. This completes the proof.


All the above theorems and discussions lead to the following result. Let X1, ..., Xn be i.i.d. spherically symmetric random vectors with X = Ri Ui, 1 _< i <_ n, where 14- " U(Sp) is independent of Ri. Then h*(Xl), ..., h*(Xn) are i.i.d. spherically symmetric random vectors with h*(Xi) = R*(Ri) Uij, 1 < i < n, R*(Ri) is independent of U

EH0[h*(_)] = 0 and Var-COVH0[h (-)] =�


where R*(Ri) and T2 are defined in (5.2.11) and (5.2.8), respectively.





5.3 AsymDtotic Null Distribution of SS

In this section we establish the null limiting distribution of SS under the class of elliptically symmetric distributions, with density function as defined in (3.2.1). Since the









test based on SS is invariant with respect to nonsingular linear transformations, it suffices to consider

Xi = RiUi, 1 _ i < n, (5.3.1) where _Ui's are i.i.d. Uniform(Sp) random vectors and are independent of Ri's, i.i.d. postive random vectors. As a first step, we now seek an asymptotic approximation for the test statistic based on SS, which possesses the following two properties under H0 : (a) the difference between SS and the approximating statistic converges to zero in probability, and

(b) the limiting distribution of the approximating statistic is easily established. One candidate satisfying these considerations is SS*- 4p 1 2 c0s(aik'k'), (5.3.2) n(n+) iki'

where cik,i'k' = the angle between Xi + Xk and Xi, + Xk,. The first property of this approximating statistic is stated and proved in the next theorem. Theorem 5.3.3 If the observations Xi's are as defined in (5.3.1) and Ho is true, then SS - SS* -:-> 0 as n -- oo.


Proof of Theorem 5.3.3 In proving this result we will need to show EH0[(SS - SS* )2] -_ 0 as n --> oo. Note that, using expressions (5.1.3) and (5.3.2),

EH0[(SS - SS* )2] = 1 , [cos i nn ) iki'

COS(aik,i'k')]'[COS(9 Pst,s't') - cos(ast,s't')] I


(5.3.4)









It can be easily shown that if (i, k) = (i', k') or (s, t) = (s', t'), then the term is zero. Also, if any one of the pairs (i, k), (i', k'), (s, t) or (s', t') is disjoint from the integers in the others, then the expected value is zero. The outline of this proof is given in Appendix B. An examination of the terms in (5.3.4) shows that, the number of terms with no disjoint pair is of the order n6. This is explained as follows. Suppose pair (i, k) is not disjoint from the integers in the other pairs, then there exists a pair with at least one common integer as in (i, k). Without loss of generality, let us assume this pair is (i, k'). Now the pair (s, t) could be disjoint from pairs (i, k) or (i, k'), but must have at least one integer common with that of (s', t'). (Otherwise pair (s, t) will be disjoint.) Let us call pair (s', t') as (s, t'). Thus the number of terms with no disjoint pair is n(n+l) n(n+l) n6.
5-'n.~l Y' 1 n. n =-n
2 2
i<_ki'

Therefore we can bound (5.3.4) by

(constant).EH0t [cos(7 Piki'k') -cos(aik,i'k')]-[cos(E Pst,s't') - cos(ast,s')] }


< (constant).EH0 t 11 Pik,i'k' - O0ik,i'k'' I Pst,st' - ctst,s't'I }


Note that

11 Piki'k' - aik,i'k'l" b1t Pst,st' - aOst,s't'l < 2.2 = 4 a.e.


Furthermore, the 7t P's are consisent estimators of their respective cc's. Thus, by the Lebesgue Dominated Convergence Theorem, see, e.g., Chow and Teicher (1978), p. 99, we have

EH0 [t P~ iki'k' - Ok,i'k''[ IPst,s't' - st,s't'I } - 0 as n ----> oo.


Hence we have proved








EHO[(SS - SS* )2]


This completes the proof, since convergence in quadratic mean implies convergence in probability.


The next theorem proves the limiting distribution of SS* under H0.


Theorem 5.3.5 If the observations Xi, 1 _< i _< n, are as defined in (5.3.1) and H0 is true, then


12 4 X2 as n -- 0 4sc2;


where t2 is defined (5.2.8). Proof of Theorem 5.3.5 Note that

aik,i'k' = angle between (Xi+-Xk) and (Xi,+Xk,)

= angle between Ii+Xk and Xi+Xk 1Xi + Xk an I -'+-k Hence, we can write

cos(aikikX) i+X--k ) Xi'+Xk'
c~s(�qk~i'1' = (Ii+Xk 11 11 Xi,+.Xk, 'l since Xi+Xk _ has norm 1. Now SS* in (5.3.2) can be written as

SS* - 4p ( Xi+Xk Xi+Xk
n(n+l)2 5 (' k,11 X i+X_k 11 11. --, k -- .


4p(X Xi+Xk )T.( Xi'+Xk, n(n+1)2 i

--- 0 as n --> oo.





81


X+X_ XiT+nk,
, n__iK --k T(n+
n(n+l) IXi+Xk I n1) Xi+Xk'
2 i<_k 2 i'
p- R T (5.3.6)
pRR,


where


R- -"n Xi+Xk
R=n(n+1) IIXi+Xk 11
2 i

Moreover let us write


R4 n X-i+X24
-n= (n(n+l) 11 Xi+Xk 11
2 i

q n X_+ X~
+ n(n+l) ." I1i+2 tl)
2 i=k


n �*
(- -j) Uin + (2,,,


1I V Xi+X.k Uln-(2) iAk 11 X+Xk II


1 n X.i and U2n =---i- 1 1 Xi 11


Since, under H0, Li 1 < i :5 n, are i.i.d. Uniform (Sp) random vectors (as shown in Theorem 5.2.4), we have

P
U2n 4 0 as n - oo. n-i 2F-n
Using the fact that -- - "+1 and n+ -+ 0 as n-- oo, we see that, under H0, expression (5.3.7) implies


Rn- 'n Uln :- 0 as n -- oa.


(5.3.8)


where


(5.3.7)








Combining expressions (5.3.6) and (5.3.8), now it suffices to find the limiting distribution of NFn Uln under H0. Define

* 2 n(539) UI=-- i". h* (Xi),(539 Uln ni=l1


where h* is as defined in (5.2.5). It follows from Theorem 3.3.13 (Randles and Wolfe [1979], p. 82) that, under Ho, S(Uin- Uln) - 0 as n --- oo, and hence expression (5.3.8) implies Rn- -n1 UYn 0 as n -- co. (5.3.10) Note that, under Ho, n .



where h*(Xi)'s are i.i.d. with HOA -X01= 0Q and Var o[h* (X)] -2 (see section 5.2). Hence by the multivariate central limit theorem we have, under H0, 4-i * 4 Np(0,-'-.Ip) as n -- oo. (5.3.11) Using expressions (5.3.10) and (5.3.11), it follows from Slutsky's Theorem that, 1d4t2
R Np0, --( p) as n -* oo,











2 R 2T -


4 Np(, p) as n ->00.


Therefore, under H0, using (5.3.6), we have


- (2 Rn)T(-ERn)
2-n 2,r


42
- 2 as n -4 oo.


This completes the proof.


Now we are ready to state the theorem for the limiting distribution of SS under H0. Theorem 5.3.12 If the observations Xj, 1 < i < n, are as defined in (5.3.1) and Ho is true, then


SS d) 2 -:4, xp as n --4 0o
4ss2 4P2


where t2 is defined (5.2.8). Proof of Theorem 5.3.12 With Theorems 5.3.3 and 5.3.5 established, the result follows directly from the Slutsky's Theorem.



Note that when p = 1, cr2 = 1/3. Thus when p = 1, SS 3 .__Sd42 4,t2 4 X as n - oo. Finally, to perform the test will require a consistent estimator of C2, say 2. The test will


then be based on the fact that under H0,


and hence


SS*
4,c2








S S 4 X 2 as n --> oo"

4 2

For consistent estimator of c2, we consider


= 2 n
n(n-1)(n-2)


(5.3.13)


I" COS( Piki
k

k k' i

where Pikik' is as defined in (5.1.4). The consistency is proved in the next theorem.



Theorem 5.3.14 Let t2 be as defined in (5.2.8). Then t2 is a consistent estimator of C2 under H0.

Proof of Theorem 5.3.14 Define


"2 = 2 n
n(n-1)(n-2)


kcos(aik,ik'). k

k k' i


Using the same arguments as in the proof of theorem 5.3.3, it can seen that under H0,

_2 -:" 0 as n -- oo. So we can conclude our proof by showing, under H0, E -t2 --0as n --- oo. Note that

"r2 = EHO[(h*(X1))Th*(X-)] (by expression (5.2.8)) (XI+X2)T(XI+X3)
- H0 II X1+ 1 X4+~i J(by expressions (5.2.3) and (5.2.8))








H (Xi+Xk)T (xi+Xk)
- EH0[LI i+X II i+Xi


], 1

= EH0[g(Xi, Xk, Xk*)],


(5.3.15)


where

(Xi+Xk)T(Xi+Xk,)
11 Xi+Xk 11 11 Xi+Xk' II satisfies g(Xi, Xk, Xk,) = g Xk)- Also we can write


j2 2
n(n-1)(n-2) i=l


2 n
- n(n-1)(n-2).I


I cos(aik,ik,) k k k' i


I-


1=1 K k k'*i


(Xi+Xk)T(Xi+Xk,) 11 Xi+Xk 11 11 Qi+24' 1


2 1 1rf (Xi+Xk)T(--i+X--k')
-n(n-1)(n-2) i

(X+Xi)T(X +Xk,) + 11 _k+ -x i 1111 X kc+ Xgk' 11


(-k'+Xi) T(-+Xk)
+ II Xk,+Xi 11X1k+Xk I1


f (Xi+Xk)T(xi+Xk )
3() i

Xk+Xilii)T (k+Xk,)
+I2L.k+Xi 111 IIKfk+_-.k' I


(Xk'+Xi)T(Xk'+Xk)
+ 11 Xk'+Xi ilt Xk,+Xk II


1 9-x *(Xi, Xk, Xk'),
(3) i

*(X Xk, &') = 1 { (Xi+Xk)T(i+Xk,) i+ (&+Xi)T(Xk+Xk,)
3 11 Xi+Xk 1111 Xi+Xk, II II Lk+Xi 11 k+_k, II


where









(-X,+Xi)T(xk,+Xk) (5.3.16) + TI TX. liX i n Xk,+Xk I



3 (-Xi, Xk, Xk) + g (Xk, Xi, Xk')+ 9(-Xk', Xi, Xk) I


is symmetric in its arguments. Thus, by Corollary 3.2.5, Randles and Wolfe (1979), p. 71, we have, under H0,


,[2-C2 - 0 as n -4 oo.







5.4 Asymptotic Distribution of SS under Contiguous Alternatives

In this section we establish the asymptotic distribution of SS under a sequence of alternatives approaching the null hypothesis H0 : = 0 for a specific class of elliptically symmetric distributions. As a first step, let us assume X1, ..., Xn are i.i.d. as X = (X1 ..., Xp)T, where Xis elliptically symmetric with a density function fx of the form


fx x= K p x_.'e/2exp{-[e-0)T'lx-o*)/COlV}, xE RP, (5.4.1) where Kp, CO are as defined in (2.3.2), e is the point of symmetry, and Z is the variancecovariance matrix. Since both SS and Hotelling's T2 are affine-invariant, we can, without loss of generality, assume that . = Ip, the pxp identity matrix. Thus, under H0, X1 ..., Xn are i.i.d. with density of the form


fx(&) = Kp.exp(-[(_Tx)/Co]V}.


(5.4.2)









Under the sequence of alternatives considered in section 2.3, X1, ..., Xn are i.i.d. with density fx(x-cn-1/2), where fx is given in (5.4.2) and c E RP-{0) is arbitrary, but fixed. Substituting p+1 for p in the proof of Appendix A, we have shown that if 4v + p > 2, then

0 < Ic(fx)= f [ J x ] fx()dx < o for all c # 0. (5.4.3) Jfx&) _Thus, when 4v + p > 2, the previous results in conjunction with LeCam's Theorems on contiguity (see Hijek and Sidd.k [1967] pages 212-213) can be used to establish the asymptotic distribution of SS under the sequence of contiguous alternatives. This is stated in the next theorem.


Theorem 5.4.4 Let X, .... Xn be i.i.d. from an elliptically symmetric distribution with density function fx given in (5.4.1) with I= Ip, the pxp identity matrix. If 4v + p > 2, then under the sequence of contiguous alternatives for which X has density of the form fx(Z-cn'1/2), defined in (5.4.2), we have
SS ... 2 4v 2 2 -2
SS - ( C4v {E [R2V-R*(R)] } 2c) as n -- o, (5.4.5) z" pcH



where t2 and R*(R) are as defined in (5.2.8) and (5.2.11), respectively. Proof of Theorem 5.4.4 Define I cn T_ fX(_.i)
Tn- = I
Ni=lI f (-i


* 2 n �
Uln = n" Y C (--i),
i=l





88

N~T *
Sn k U=n,


where c, (= RP- {0}, and h* is defined in (5.2.5). Here Xi = Ri Ui, where U. is distributed uniformly on the p-dimensional unit-sphere independent of Ri, and R = x TX. thus we can write

Tn - 2v i _j
i--1

_ n T 2vlU Y _ cTi2 -1.
C vFn i=1


Thus, under H0,



T -5 i__ I --- _ri "0


i 2XTTR*(Ri) Ui

-n i=1 - c R 2v .i



(since h*(-Xi) = R*(Ri) Ui, as proved in Theorem 5.2.9) = 1 n [ Ai (say),



where

Ai= 2TR*(Ri) Ui and Bi = 2V qTR.2v- U. Note that, under HO, [BJ's are i.i.d. with









E[:j o and VaA.=[ 011 o12d 1 012 02 2where
2)
Oll= E(Ai)


= E(2 TR*(Ri) Ui .2R*(Ri) uiT)

4E{ [R*(Ri)]} , E(Ui-)

4 E{ [R (Ri)] }2.eTL
P

022 = E(B2)



= 4v2 E(R4V-2).TE(T 2v p0
p42 E(Ri 4v2)'-�-,



and

Y12 = E(AiBi)

= E(22TR*(Ri) Ui. 2v R.2v1 UiTc) CO

= 4v E(Ri2VIR*(Ri)).2,E(ij )c


=4_E[Ri.2V.1R,(Ri)].hec.
PC









Thus, under H0, see, e.g., Serfling(1980), Theorem B, p. 28,


[Sn
[T


4N2


Gl 121 ) as n - oo, 012 02


where 011, 022, and 012 are as defined above. Applying LeCam's Theorems on contiguity (see Hijek and Siddk [1967], p. 212-213), if 4v + p > 2, then we have, under the sequence of alternatives,

d
Sn 4*N(r12,Tj1)asn--4oo. That is, under the sequence of alternatives,


SN( 4V E[Ri2vIR*(Ri)].2Tc' . kT
"pC0 -P-


as n --- 00


since


E [ [R*(Ri)] }2 =.r2. Therefore, under the sequence of alternatives,


* d N(4v 2VR
N1ii Lin 4 pk E[R2-R'(i],


4T)


as n -4 00,


and hence


-F * _-d Np( 2 E[Ri2V'R*(Ri)].c, IP) as n -* oo.


Recall that


* = RT R - -Fn Un -:4 0as n--> oo and SS* =p -R~




Full Text

PAGE 1

INTERDIRECTION TESTS FOR REPEATED MEASURES AND ONE-SAMPLE MULTIVARIATE LOCATION PROBLEMS BY SHOW-LI JAN 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

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Copyright 1991 by Show-Li Jan

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To my mother and to the memory of my father

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ACKNOWLEDGEMENTS I am very thankful to Dr. Ronald H. Randles, without whom this dissertation would not have been possible, for being my dissertation advisor. His encouragement, helpful comments, kindness, ideas, and invaluable assistance are important contributions to the development of this dissertation. Very special thanks are due Dr. Malay Ghosh, Dr. Pejaver V. Rao, Dr. Jane F. Pendergast, and Dr. Louis S. Block for serving on my committee and for many useful suggestions. I also thank Dr. Michael Conlon and Mr. Rodger E. Hendricks for many helpful comments on computing using fortran and IMSL subroutines. The continual support of my family, my parents-in-law, and my friends Ms. Taipau Chia and Mr. Shanshin Ton is acknowledged with appreciation. Finally, most important of all, I would like to express deep gratitude to my husband, Gwowen, for his excellent typing and encouragement, without which this dissertation would not have been completed. IV

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TABLE OF CONTENTS ACKNOWLEDGEMENTS iv LIST OF TABLES vii ABSTRACT vxi CHAPTERS 1 INTRODUCTION 1 2 A MULTIVARIATE SIGN TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS 14 2. 1 Definition of the Test Statistic 14 2.2 Null Distribution of V n 17 2.3 Asymptotic Distribution of V n under Contiguous Alternatives 19 2.4 The Pitman Asymptotic Relative Efficiency of V n Relative to HotellingÂ’s T 2 24 3 A MULTIVARIATE SIGNED-RANK TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS 33 3 . 1 Definition of the Test Statistic 33 3.2 Asymptotics 34 3 . 3 Numerical Evaluation of ARE(nW n , T 2 ) 41 4 MONTE CARLO STUDY 47 5 A MULTIVARIATE SIGNED SUM TEST BASED ON INTERDIRECTIONS FOR THE ONE-SAMPLE LOCATION PROBLEM 66 5 . 1 Definition of the Test Statistic 66 5.2 Some Intermediate Results 70 5.3 Asymptotic Null Distribution of SS 77 5.4 Asymptotic Distribution of SS under Contiguous Alternatives 86 5.5 Numerical Evaluation of ARE(SS/4x 2 , T 2 ) 92 APPENDICES A 98 B 105 v

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BIBLIOGRAPHY 107 BIOGRAPHICAL SKETCH Ill vi

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LIST OF TABLES 2.1 ARE(V n , T 2 ) 32 3. 1 ARE(nW n , T 2 ), WITH ARE(V n , T 2 ) IN PARENTHESES 45 3.2 ERROR ESTIMATE ERREST OF ARE(nW n , T 2 ) 46 4. 1 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL DISTRIBUTION WITH E = I4 53 4.2 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL DISTRIBUTION WITH E = E E T 54 4.3 MONTE CARLO RESULTS FOR ELLIPTICALLY SYMMETRIC DISTRIBUTIONS WITH DENSITY OF THE FORM GIVEN IN (2.3.1) WITHE =U 55 4.4 MONTE CARLO RESULTS FOR ELLIPTICALLY SYMMETRIC DISTRIBUTIONS WITH DENSITY OF THE FORM GIVEN IN (2.3.1) WITHE = EE t 57 4.5 MONTE CARLO RESULTS FOR PEARSON TYPE II DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY OF THE FORM GIVEN IN (4. 1 .3) WITH E = I4 59 4.6 MONTE CARLO RESULTS FOR PEARSON TYPE II DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY OF THE FORM GIVEN IN (4. 1 .3) WITH E = E E T 60 4.7 MONTE CARLO RESULTS FOR PEARSON TYPE VII DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY OF THE FORM GIVEN IN (4. 1.4) WITHE = 14 61 4.8 MONTE CARLO RESULTS FOR PEARSON TYPE VII DISTRIBUTION WITH SHAPE PARAMETER m, AND DENSITY OF THE FORM GIVEN IN (4. 1 .4) WITH E = E E T 62 4.9 MONTE CARLO RESULTS FOR QUADRIVARIATE NORMAL MIXTURES WITH MEAN ji AND VARIANCE-COVARIANCE MATRIX E GIVEN IN (4.1.5) 63 5.1 ARE(SS/4t 2 , T 2 ) 96 vu

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5.2 ERROR ESTIMATE ERREST OF ARE(SS/4x 2 , T 2 ) 97 viii

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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 INTERDIRECTION TESTS FOR REPEATED MEASURES AND ONE-SAMPLE MULTIVARIATE LOCATION PROBLEMS BY SHOW-LI JAN March, 1991 Chairman: Dr. Ronald H. Randles Major Department: Statistics Affine invariant interdirection tests are proposed for a repeated measures problem. The test statistics proposed are applications of the one-sample interdirection sign test and interdirection signed-rank test to a repeated measurement setting. The interdirection sign test has a small sample distribution-free property and includes the two-sided univariate sign test and BlumenÂ’s bivariate sign test as special cases. The interdirection signed-rank test includes the two-sided univariate Wilcoxon signed-rank test as a special case. The asymptotic null distributions of the proposed statistics are obtained for the class of elliptically symmetric distributions. In addition, the asymptotic distributions of the proposed statistics under certain contiguous alternatives are obtained for elliptically symmetric distributions with a certain density function form. Comparisons are made between the proposed statistics and several competitors via Pitman asymptotic relative efficiencies and Monte Carlo studies. The interdirection tests proposed appear to be robust. The sign test performs better than the other competitors when the underlying distribution is heavy-tailed or skewed. For normal to light-tailed distributions, the HotellingÂ’s T 2 and signed-rank test have good powers when the variance-covariance structure of the IX

PAGE 10

underlying distribution is non H-type, otherwise ANOVA F and the rank transformation test RT perform better than the others. An alternative test for the one-sample multivariate location problem is also proposed which extends the univariate signed-rank test to multivariate settings. The test proposed is somewhat like applying the interdirection sign test to the sums of pairs of observed vectors. It includes the two-sided univariate Wilcoxon signed-rank test as a special case. The asymptotic distributions of the proposed statistic under the null hypothesis and under certain contiguous alternatives are obtained for a class of elliptically symmetric distributions. Comparisons are made between the proposed statistic and HotellingÂ’s T 2 via Pitman asymptotic relative efficiencies. The signed sum test proposed performs better than HotellingÂ’s T 2 when the underlying distribution is heavy-tailed. However, for normal to light-tailed distributions, the HotellingÂ’s T 2 performs slightly better than the proposed test x

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CHAPTER 1 INTRODUCTION In this dissertation we investigate test statistics for certain repeated-measures and one-sample multivariate location problems. For the repeated-measures problem, we let X!, .... Y n be independently and identically distributed as Y , where Y = (Y^ Y p ) T is from a p-dimensional, p > 2, absolutely continuous population. For each subject i, we shall regard Yj as repeated measures with one observation for each of the p treatments. We use the general mixed model with a subject by treatment interaction. In vector form, we consider the model (see, e.g., Winer [1971], p. 278), Y i = p i l p +i + fii i +£ i ,i= 1, ..., n, (1.1.1) where I p is the pxl vector of l’s, 1 = (Xj, ..., x p ) T is the vector of fixed treatment effects, and random variables (3j, £lj, £j for i = 1, ..., n are all mutually independent. (More details about this model will be given in section 2.1.) Note that the variance-covariance matrix of fili is probably not H-type, to be described later. We are concerned with the test of equal treatment effects, which in model (2.1.1) can be described as: H 0 : Tj =x 2 = ••• =T p versus H a : Xj * Xj' for some j * ]. (1.1.2) We first consider several parametric statistics for this problem when the population is p-variate normal. Probably the most well-known parametric procedure for this problem is based on Hotelling (1931) T^ test. Define the (p-l)-variate random vector Zj by — i (Zjj, .... Zj p .j) (Yj! Yj p , .... Yj p .2-Yi p ) T , i = 1 , .... n. 1

PAGE 12

2 The test of (1.1.2) can be carried out by the Hotelling’s T 2 statistic computed from the mean vector and sample variance-covariance matrix of the vectors of differences Zj’s. This is due to Hsu (1938). The Hotelling’s T 2 is then defined as T 2 = T 2 ( Z h ..., Z n ) = n Z 7 (I)' 1 !, (1.1.3) where 2 Z Zi, and £ = -py £ (Zi-Z)^-!) 7 . i=l 11 i=l Under Hq, T is asymptotically Chi-square with p-1 degrees of freedom. If the underlying population is p-variate normal, then the null distribution of T 2 is a multiple of the F-distribution with p-1 numerator degrees of freedom and n-p+1 denominator degrees of r\ freedom. Hotelling’s T is invariant with respect to nonsingular linear transformations of the observations Zj, i = 1, ..., n. That is, if D is any nonsingular (p-l)x(p-l) matrix, then T 2 (DZ 1 ,...,DZ n )=T 2 (Z 1 ,...,Z n ). (1.1.4) We shall call this invariance property affine-invariance. This appealing invariance property ensures that the value of the test statistic remains unchanged following rotations of the observations about the origin, reflections of the observations about a (p-2)-dimensional hyperplane through the origin, or changes in scale. Hence the performance of Hotelling’s T test will not depend on the structure of the population variance-covariance matrix or the direction of shift. Another parametric procedure for this problem is the classic ANOVA F test. The test is based on the original observations Yjj, 1 < i < n, 1 < j < p, and is defined as P _ _ F = n I (Y.j-Y y i=l iTT i, £ (%Yi.-Y. j+ Y. ,) 2 1=1 J=1 (1.1.5)

PAGE 13

3 where Y ; = £ Y; n i=l •J U’ IPIP = Z Y and Yj = £ Yj:. If the underlying population is P j=l ' J ‘ P j=i J p-variate normal with variance-covariance matrix of the form 1 = ( 1 . 1 . 6 ) then the null distribution of F is F-distribution with p1 numerator degrees of freedom and (n-l)(p-l) denominator degrees of freedom. This test for (1.1.2) under the variancecovariance matrix (1.1.6) was obtained by Wilks (1946) from the generalized likelihood ratio principle when the underlying population is p-variate normal. The matrix in (1.1.6) is said to have compound symmetry. While compound symmetry is a sufficient condition for the test statistic F to have an exact F-distribution, it is not a necessary one. Huynh and Feldt (1970) have found a necessary and sufficient condition, which may be expressed in three alternative forms (see Morrison [1976], p.152): (1) The population variance-covariance matrix L = (a-,), 1 < j,j' < p, has the pattern defined by JJ aj+aj-U if j = j' OCj-HXj. if j * j' (1.1.7) where X, a h ..., a p are (p+1) arbitrary constants such that the resultant matrix is positive definite. (2) All possible differences Yj-Yj 1 (j ^ j') of the response variates have the same variance. (3) Define e, a function of the elements of Z, by

PAGE 14

4 P 2 (°d
PAGE 15

5 Under H 0 , the test S is asymptotically Chi-square with p-1 degrees of freedom. (See, e.g., Hollander and Wolfe [1973], p.140.) Define (n-l)S _ n(p-l)-S • Then under H 0 , the test F s is compared with an F-distribution with degrees of freeedom p-1 and (n-l)(p-l). A more accurate approximation to Friedman’s test is proposed by Jensen (1977). Iman and Davenport (1980) proposed two new approximations and also pointed out that the F approximation is better than the % 2 approximation. Another rank test was proposed by Koch (1969). His test statistic used the ranks of the aligned observations, obtained by subtracting from each observation the average of the observations for that block. This alignment process will eliminate or at least reduce the block effect. To introduce his test, let us define Rjj = rank of (Yy-Yj) among Y n -Y L , ..., Y np -Y n, R. J = i| 1 R ii andR i. = f.| R ijThen the test statistic has the form nf (Rj n -^ 1 ) 2 W‘ = EL_1 i 1 n p n(p-l) £ £ ^1J d \2 I X (Rij Ri.) ( 1 . 1 . 10 ) If Yj, ..., Y n are i.i.d., then under the null hypothesis of the exchangeability of the components of Yj = (Yjj, ...,Yj p ) T , the test W has an asymptotic Chi-square distribution with degrees of freedom p1 . Unlike the Friedman test, which depends entirely on within-block rankings, Quade (1979) considered a p-sample extension of the Wilcoxon signed rank test by taking

PAGE 16

6 advantage of the between-block information. This is done by considering weights assigned to each block on the basis of some measure of within-block sample variation, such as the range, standard deviation, mean deviation, or interquartile difference. To illustrate this test statistic, let us define Dj = D(Yj), a location-free statistic that measures the variability within the ith block, Qi = rank of D, among D b . . ., D n , and Ry=rank of Yy among Y n , ..., Y ip . Quade's procedure, based on weighted within-block ranks, is defined as W P r n -1 r\ 72l[lQ i R ij ] 2 j=l i=l p(p+l)n(n+l)(2n+l) 9(p+l)n(n+l) 2(2n+l) 72 i [ £ QifRij-^)] 2 1=1 i=l p(p+l)n(n+l)(2n+l) ‘ ( 1 . 1 . 11 ) Under the null hypothesis of the exchangeability of the components of Yj, the test W has an asymptotic Chi-square distribution with p-1 degrees of freedom. A natural nonparametric analog of ANOVA F test was proposed by Iman, Hora and Conover (1984). Their procedure is first to transform all the observations to ranks from 1 to np and then apply the parametric ANOVA F test to the ranks. This approach retains both the withinand between-block information. Defining Rjj = rank of Yy among Y n , ..., Y np , the rank transformation test is defined as

PAGE 17

7 RT = nf (R.-R Y _ >1 n-1 .? ;( R ij R iR .j +R ..) 2 i=l j=l ( 1 . 1 . 12 ) 1 " 1 P 1 P where R j n X Rij, Ri. = ~ Z Rij and R = Z R.jAssuming Yy's are mutually independent, and considering H 0 : F ij = Fifor i = 1, .... nandj = 1, ..., p, where Fy is the distribution function of Yy, then the asymptotic null distribution of (p-l)RT is Chi-square with p-1 degrees of freedom under suitable conditions. Their simulations showed that the behavior of test RT is closely approximated by the F-distribution with (p-1) and (n-l)(p-l) degrees of freedom. Comparisons made among ANOVA F test, Friedman’s test (S), Quade's test (W), and rank transformation test (RT) via Monte Carlo studies showed that the F test had the most power for normal distributions, the Quade's test and F test were almost equivalent and gave the best results for uniform distribution, the Friedman's test and the RT test gave similar results and were best for the Cauchy distribution, and the RT test has the most power for double exponential and lognormal distributions. Hora and Iman (1988) developed the limiting noncentrality parameters of the rank transformation statistic and some other tests, which were then evaluated to make comparisons among those tests via Pitman asymptotic relative efficiencies. Agresti and Pendergast (1986) also considered a test that is appropriate when the null hypothesis (1.1.2) is expressed as the exchangeability of the components of Yp Their procedure utilizes a single ranking of the entire sample. Let Rij = rank of Yy among Y n , ..., Y np , Rj = i £ R ijt

PAGE 18

8 p = Corr(Rjj, Rjj-), j * j', and ct 2 = Var(Rjj). The test statistic is based on (1.1.13) o (1-P) This test includes Koch's test and the rank transformation test as special cases. They argued that under Hq their statistic has an asymptotic Chi-square distribution with p-1 degrees of freedom if the asymptotic distribution of R = (R A , ..., R p^) 7 is (p-l)-variate normal. Since they did not present conditions guaranteeing this normality, Kepner and Robinson (1988) concluded the work of Agresti and Pendergast by determining reasonably sufficient conditions for R to have a (p-l)-variate normal limiting distribution. In analogy to the Iman, Hora and Conover (1984) proposal of a rank transformed version of ANOVA F test, Agresti and Pendergast (1986) considered a rank transformed version of Hotelling's test. This procedure is appropriate when the hypothesis of no treatment effects is more broadly expressed as the marginal homogeneity condition F 1 = f 2 = ••• = Fp> where Fj, ..., F p denote the one-dimensional marginal distribution of *T* Y = ( Y j , . . . , Y p ) . Their statistic is based on RT h = n R T S _1 R, (1.1.14) where i=l Their simulations showed treating the null distribution of RT H as a multiple of Fdistribution with p-1 numerator degrees of freedom and n-p+1 denominator degrees of freedom is a reasonable approximation. They also argued that under H 0 , their statistic has

PAGE 19

9 an asymptotically Chi-square distribution with p1 degrees of freedom if the asymptotic distribution of (R j, R p _i) T is (p-l)-variate normal. Their simulations showed that the RT and RT H statistics behaved much like their parametric analogs. We now consider the multivariate tests for the one-sample location problem. As a first step, we let Xj, ..., X n be i.i.d. as X = (Xj, ..., X p ) T , where X is from a p-variate absolutely continuous population with location parameter §* (pxl). We would like to test Ho:£* = 0 versus H a :£**0. (1.1.15) Here 0 is used without loss of generality, since H 0 : 0* = 0Q can be tested by subtracting 00 from each observation X; and testing whether these differences (X; GqI’s are located at Q. The classical procedure used in this setting is Hotelling’s T 2 , which is defined as T 2 = T 2 ( X h . . ., X n ) = n X 1 ©-^, where X=-^-ix i; and£=-p r I(X i -X)(X i -X) T . If the underlying population is p-variate normal, then the null distribution of T 2 is a multiple of the F-distribution with p numerator degrees of freedom and n-p denominator degrees of freedom. The affine-invariant property of Hotelling’s T 2 is discussed earlier on page 2. Many nonparametric procedures have been proposed. The most popular statistic is the component sign test, which is a nonparametric analog of Hotelling’s T 2 using signs of Xjj’s, 1 < i < n, 1 < j < p. (See, e.g., Randles [1989], p. 1045.) Let Sj = X sgn(Xij), and£= (S b ..., S p ) T , i=l

PAGE 20

10 where sgn(x) = 1(0, -1) for x >(=, <) 0. The test is based on S* = S. T ( n$f )' 1 S, (1.1.16) where $/= (wjj-), wjj=-jpI sgn(X i j)sgn(X ij O, for 1
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11 transformations have little effect on the small sample power of the tests for nearly degenerate distributions. Two well-known bivariate sign tests are due to Hodges (1955) and Blumen (1958). Their procedures are affine-invariant and have a distribution-free property. Joffe and Klotz (1962) presented an expression for the exact null distribution of the Hodges bivariate sign test. They also computed the Bahadur limiting efficiency of the test relative to the Hotelling’s T 2 test for normal alternatives. Killeen and Hettmansperger (1972) made an exact Bahadur efficiency comparison of Hotelling’s T 2 with respect to both Hodges’ and Blumen's bivariate sign tests. Klotz (1964) obtained exact power for the bivariate sign tests of Hodges and Blumen under normal alternatives and therefore permitted comparisons of the two tests for sample sizes n = 8 through 12. The procedure proposed by Bennett (1964) for the bivariate case is a signed-rank test generalizing Wilcoxon's univariate signed-rank test. This test is not affine-invariant. Another affine-invariant bivariate rank test was introduced by Brown and Hettmansperger (1985). Their statistic is based on the gradient of Oja's measure of scatter (Oja, 1983). Letting A(Xj, Xj; £ ) denote the area of the triangle formed with Xj, Xj and £* as vertices, define T(fi*) = ZIA(X i ,X j ;fi*). (1.1.17) i
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12 where Q 2 n (-Xi) is computed using observations Xj, X n , and their projections through the origin, -Xj, -X n . Their test for (1.1.15) is defined as S 7 ^' 1 ^. Under Hq, the test statistic is asymptotically Chi-square with 2 degrees of freedom. Oja and Nyblom (1989) also studied the bivariate location problem. Their tests are analogs of the univariate sign test. Denote the direction angle of Xj by 9,. Then 6* = 0J+7U / r / (mod 27t) is the direction angle of -Xj. Write < 0 2 < ... < 0 2n for the ordered angles in the set {0j, ..., 0 n , 0 x , .... 0 n ). Define Zj = 1 ifO'e {0j 0 n ) 0 if0je {0*, ..., 0*},i=l, ..., n, (1.1.19) and Zn +i = 1-Zj , i =1,2,.... (1.1.20) T* The vector Z = (Zj, ..., 7 ^) indicates which of the observations lie above or below the horizontal axis. They proposed using test statistics based on Z. The test statistics are distribution-free and affine-invariant, and include Hodges (1955) and Blumen (1958) sign tests as special cases. Also, they proposed some new intuitively appealing tests. A general class of these invariant sign test statistics is Z [£ (Z k+i -Hh(i/n)] 2 ( 1 . 1 . 21 ) k=0 i=l z where h is a suitably chosen score function. In the recent study Randles (1989) proposed an interdirection sign test for this problem. His test statistic included the two-sided univariate sign test and the Blumen (1958) bivariate sign test as special cases. Also, Peters and Randles (1990) suggested a signed-rank test based on interdirections, which includes the two-sided Wilcoxon signed-

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13 rank test as a special case. The bivariate case of their statistic was considered in the dissertation of Peters (1988). The interdirection sign test and the interdirection signed-rank test will be described in detail in Chapters 2 and 3, respectively, where they are applied to a repeated measures problem. In this dissertation, the interdirection sign test for a repeated measures problem is defined in Chapter 2. The asymptotic distributions of the test under H 0 and under certain contiguous alternatives are obtained in sections 2.2 and 2.3, respectively. The Pitman asymptotic relative efficiencies of the test relative to HotellingÂ’s T 2 are presented in the last section. In Chapter 3, the interdirection signed-rank test for the same problem is described, with its asymptotic distributions obtained in section 3.2 and the evaluations of the ARE of the signed-rank test relative to HotellingÂ’s T 2 established in the section 3.3. Comparisons of several competing procedures are made in Chapter 4 via Monte Carlo studies. An alternative test for the one-sample multivariate location problem is proposed in Chapter 5. Some useful intermediate results are presented in section 5.2. The asymptotic distributions of the test under Hq and under certain contiguous alternatives are developed in sections 5.3 and 5.4, respectively. Finally, in section 5.5, we evaluate the ARE of the proposed test relative to HotellingÂ’s T 2 .

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CHAPTER 2 A MULTIVARIATE SIGN TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS 2, 1 Definition of the Test Statistic The multivariate sign test based on interdirections, denoted by V n , was proposed by Randles for the one-sample multivariate location problem. In this section we will show how this test statistic can also be applied to repeated-measures designs for detecting treatment effects. For the one-sample multivariate location problem, we let X lf ..., Xn, where X { = (Xji, •••, X ip ) T , be independent and identically distributed (i.i.d.) as X = (X^ ..., X p ) T , where X is from a p-dimensional absolutely continuous population with location parameter $ Q (pxl). We would like to test Ho : 0* = 0 versus H a : 0* * 0. Here 0 is used without loss of generality, since Hq : 9 = 0q can be tested by subtracting 00 from each observation Xj and testing whether these differences (Xj 0q)’s are located at Q. For the problem of single-factor repeated-measures designs, we let Yj, ..., Y n be i.i.d. as Y = (Yj, ..., Y p )^, where Y is from a p-dimensional, p > 2, absolutely continuous population. Note that the components of Yj are repeated measurements of the ith experimental unit. We will use the general mixed model with a subject by treatment interaction. In vector forms, we consider the model (see, e.g., Winer [1971], p. 278), Xj = Pil p + 1 + fili + £j, i = 1, ..., n, (2.1.1) 14

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15 where l p is the pxl vector of l’s, l = (ij, x p ) T is the vector of fixed treatment effects. Pi represents the random effect of the ith subject, Pij denotes the vector of the ith subject by treatments interactions, and e, is the vector of random error of the ith subject. We assume pj’s are i.i.d. with mean 0, j3i[’s are i.i.d. with mean 0 and a general variancecovariance matrix possibly not H-type, described in Chapter 1, gj’s are i.i.d. with mean 0 and variance-covariance matrix a e I p , where I p is the pxp identity matrix. The random variables pj, Pi,, £, for i = 1, ..., n are all mutually independent. We are concerned with the test of equal treatment effects, which in model (2.1.1) can be described as: Hq : Xi = X 2 = ... = x p versus H a : Xj * Xj' for some j j . (2.1.2) Note that the sample Yj,..., Y n has location parameter i. The problem is to test whether the components of the location parameter j are all equal. We can transform this problem to the standard one-sample multivariate location problem, described earlier, by looking at the differences among the components Y;j within each observation Yj. The transformation is described below. Define Yjl Yi 2 -Y ip -1 -1 -i Yii Y i2 Yj p-i Y ip = A Yj , i = 1, ..., n, and " Xi-tp " *2-^p i -r x 2 • = i -i ’ i -i • -Vi“V VI -^p J

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16 where 1 A = -1 -1 -i (2.1.3) is a (p-l)xp matrix. Now, we have a (p-l)-variate sample Z t , .... Z^, which can be modeled via Z ; = 0 + +8* , i = 1, ..., n. where 0 ((p-l)xl) is the location parameter. Note that the variance-covariance matrix of * , pij is possibly not H-type. Thus, testing the hypotheses given in (2.1.2) is equivalent to testing Hq:S = Q versus H a :0*O. (2.1.4) We have shown that the test of (2.1.2) based on p-variate observations Yj, ..., Y n can be earned out by using a multivariate location test based on a statistic like the interdirection sign statistic V n , computed on the transformed (p-l)-variate observations Z h ..., Z for testing (2.1.4). Now we will describe the test that rejects H 0 : 0 = 0 for large values of the statistic V n I Icosfrp^), (2.1.5) i=l k=l where ifi*k 0 if i = k, ( 2 . 1 . 6 )

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17 *[(£)-($]• and Cj^ denotes the number of hyperplanes formed by the origin 0 and other p-2 observations (excluding Z, and Zk) such that Zj and Z^ are on opposite sides of the hyperplane formed. The counts {C^ I 1 < i < k < n}, called interdirections, are used via 7ip ik to measure the angular distance between Zj and Z k relative to the positions of the other observations. This statistic, V n , includes Blumen’s bivariate sign test and the 2-sided univariate sign test as special cases. Also, it is affine-invariant and has a distribution-free property under Hq, for a broad class of distributions, called distributions with elliptical directions, which includes all elliptically symmetric populations and many skewed populations as well. In the next section, we will concentrate on the family of elliptically symmetric populations. 2.2 Null Distribution of V n In this section we will find the null distribution of V n under the class of elliptically symmetric distributions, which is defined below. Definition 2.2.1 Assuming the existence of a density function, the mxl random vector X is said to have an elliptically symmetric distribution with parameters ji (mxl) and Z (mxm) if its density function is of the form f x (x) = K m IZr 1/2 h[(x fctiVfx U)], (2.2.2) for some non-negative real-valued function h, where Z is positive definite and K m is a scalar. We will write this distribution of X as E m (ji, Z).

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18 Throughout this chapter, we will use Yj, Y n to denote the original sample, and use Zj, Z n , defined in (2.1.3), to denote the transformed sample, to which the test statistic V n is applied. Now, let’s assume Yj, ..., Y n are i.i.d. as Y = (Yj, Y p ) T , where Y is E p (ji, Z). To apply the result of null distribution of V n under elliptically symmetric distributions, proved by Randles, we shall first prove that the transformed sample Z \ , .... Z n is also elliptically symmetric. To do this, we use the following lemma, which was given as an exercise in Muirhead’s (1982) book. Lemma 2.2.3 If X is E m (y., Z) then : •jfTy (i) the characteristic function x(t) = E(e — ) has the form • T $x(t) = e 1 ^ y(t T Zt) for some function \j /, (2.2.4) and (ii) provided they exist, E(X) = g and Cov(Y) = aS for some constant a. Theorem 2.2,5 If Y_ is E p (ii, Zj and Z = A Y, defined in (2.1.3), then Z is E p .i(A y., A Z A t ). Proof of Theorem 2.2.5 Since Y is Ep(ji, Z), by Lemma 2.2.3, the characteristic function of Y at I, a pxl vector, has the form of • T x(i) = ^ V(l T S t) for some function \\ f. Thus, the characteristic function of Z at s, a (p-l)xl vector, is bz(s) = E(e i T — ) = E(e i T — — ) = E(e i ^ T ^) = [e i ^ T ^]-v K [(A T ^ T Z(A T ^]

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19 = [e isT( ^-^]-\|/[s T (ALA T )s]. Thus, Z is Ep.^A ju, A Z A 7 ). We are now prepared to state the following theorem. Theorem 2.2.6 Assume the observations Yj, ..., Y n are i.i.d. from a p-variate elliptically symmetric distribution. Then, under H 0 , defined in (2.1.4), V n , defined in (2.1.5), computed on observations Zj, ..., Z n , has a small-sample distribution-free property and a 2 limiting Xp^ distribution. Proof of Theorem 2.2.6 See Randles (1989), p. 1046-1047. 2,3 Asymptotic Distribution of V n under Contiguous Alternatives In this section we will find the asymptotic distribution of V n under a sequence of alternatives approaching the null hypothesis Hq : 0 = 0. In doing this, we will restrict our attention to a specific class of elliptically symmetric distributions. LetÂ’s assume Y|, ..., Y n are i.i.d. as Y = (Yj, ..., Y p ) , where Y is elliptically symmetric with a density function fy of the form fy(y) = K p lir 1/2 exp{-[(y u) 7 !'^ u)/C 0 ] v }, y e R p , (2.3.1) where pf(p/2v) K vf(p/2) 0 f[(p+2)/2v] Â’ p r(p/2v)(7tC 0 ) p/2 Â’ (2.3.2) r(w) = j x w ' 1 e' x dx for w > 0, 0

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20 and R p is the Euclidean p-space. It can be verified that the expression in (2.3.1) is a valid density function and that represents the mean and E, the variance-covariance matrix. This family includes the multivariate normal distribution (v = 1), heavier-tailed distributions (0 < v < 1) and lighter-tailed distributions (v > 1). As explained in the previous sections, we will need to derive the density function of the transformed sample, whose form will be used when deriving the Pitman asymptotic relative efficiency of V n relative to Hotelling’s T in next section. Lemma 2.3.3 For the family of distributions given in expression (2.3.1), the transformed sample Zj, . . ., Z n has a density function of the form f z (z) = K p IA I A T r 1/2 g[(z-Aii) T (A Z A^fe-Au)], z e R p_1 , (2.3.4) where g(t) = J exp{-[(t+s 2 )/C 0 ] v }ds. (2.3.5) -oo Proof of Lemma 2,3.3 Letting x = then we can write yi-y P yp-i-yp . y P . y = .£l y = By, where B = ’A' .<£] T • £p = (0, ..., 0, 1) , a pxl vector with 1 on the pth component and 0’s elsewhere, and A is defined in (2.1.3). Since B is nonsingular, it follows that y = B^x. The jacobian of the transformation, denoted by J(y => x), is equal to IB' 1 ! = IBP 1 =1. Hence the density function of X is

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21 f x(x) = fy(x)l x=B _1 x = KpISI' 1/2 exp { [ (B ' 1 x-ja) T L' 1 (B ' 1 x-ji)/ C 0 ] v } , = K p l Zl’ ^exp { [ (xB yJ T (B E B T )' 1 (x-B jjl)/ C 0 ] v } . Since IB E B T I = IZI due to IBI= 1, it follows that fxOO = KplB Z B T I ' 1/2 exp { [(x-B ft) T (B Z B T )‘ 1 (x-B Ji)/Co] v } . T Letting B Z B = V and B = n, we rewrite the above expression as fx(x) = K p IVr 1/2 exp{-[(x-Ti) T V1 (x-ii)/Co] v }. Thus, the density function of Z is f z(2) = | f x00 dy p -oo oo = Kpivr 1/2 jexp{-[(x-n) T V1 (x-n)/Co] v }dyp. (2.3.6) -OO Note that V = BZB t = AZA t A Z e p ep T IA T eJZe p and n = Bu = A ji L^p a .Mp. Denoting V by Yu Y 12 Y21 V22 , we have Vu = A Z A T is nonsingular and V 22 = e 1 Z e p -pis a positive scalar. Next we use the fact that V 1 = Yu Y12 Y 21 V 22 J -rl Y21

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22 v; 1 ! + r l 1 1 Y 12 v2 ‘ 2 . 1 y 21 vj 1 1 -Yi'vnV^., -viiY 21 rA y 1 v 22 -l where V 22I = V 22 ~ Y 2 iY‘i\Yi2 We can expand the quadratic form in expression (2.3.6) as (x-n) T Y' 1 te-n) = (z-fl) T YA(s-fi) + (l-fl) T Y' 1 l 1 Yi 2 V 2 i ! . 1 Y 2 iY' 1 1 1 (i-fi) + (y p -h>) 2v 22-i ^-fflVAYiaV^yp-ty + [(y P -Mp)v»r(rf> T iiiii2^] 2 (2.3.7) Using the fact that 00 = Yi ii-iv 22 y 21 v' 1 1 y 12 i = iy u HV 22 .,i and expression (2.3.7), we may write expression (2.3.6) as fz(z) K p IYul' 1 / 2 IV 22 . 1 l ‘ 1/2 • / expt-Ite-fflVAfe-fi) + ((yp-Hp)vA' 2 , (2-ffi T Y' 1 1 1 Yi 2 V 2 1 2 A) 2 )/C 0 ] v )dy p . -oo Taking s = (yp'M-p)V 2 2 ^ (z-fi) T V / 1 Y 12 V 224 > we can write the above expression as

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23 oo f z (z) = KplV n r w j exp{-[((z-a) T V' 1 ‘ 1 (z-fi) + s 2 )/C 0 ] v )ds. (2.3.8) -OO Since Vj] = ASA and 0 = A ji, expression (2.3.8) is equivalent to expression (2.3.4). This completes the proof. It can be verified that expression (2.3.8) is valid density function. Now, we are in the position to discuss the asymptotic distribution of V n under a sequence of alternatives approaching the null distribution. Under Hq, Z \, ..., Z„ are i.i.d. with density of the form %(z) = K p IAIA T r 1/2 ~ exp { [ (z t (A Z A t )' 1 z+s 2 )/C 0 ] v } ds. (2.3.9) -OO Under a sequence of alternatives let Zj, ..., be i.i.d. with density fztz-cn' 1 ^), where fg is given in (2.3.9) and c € R p ‘ 1 -{0} is arbitrary, but fixed. It is shown, with the outline of the proof given in Appendix A, that if 4v + p > 3, then r c T df z (z) -,9 0 < I £ (fz) = f L ~f z (z) J f z(^) d 2 < 00 for all c ^ 0. Thus, when 4v + p > 3, the rationale of Hajek and Sidak (1967), p. 212-213 and earlier, shows that the alternatives are contiguous to the null hypothesis. Noting that both V n and j Hotelling’s T are affine-invariant, we can, without loss of generality, assume throughout that A ZA t = Ip_i, the (p-l)x(p-l) identity matrix. Theorem 2.3. IQ Let Yj, ..., Y n be i.i.d. with density function fy given in expression (2.3.1). Assume the density function of Zj, ..., Z n has the form of fz(z) = K p g(z T z) = K p j exp { -[(z T z+s 2 )/C 0 ] V } ds, z e R p_1 . -OO

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24 If 4v + p > 3, then under the sequence of contiguous alternatives. d ? v n — > Xp_i Rg'(R 2 ) g(R 2 ) (2.3.11) where g(R 2 ) = j exp{-[(R 2 +s 2 )/C 0 ] v }ds, -oo and g'(R 2 ) = dg(R 2 )/dR 2 and R 2 = Z T Z. Proof of Theorem 2.3.10 See Randles (1989), p. 1050. Note that the noncentrality parameter in (2.3.1 1) can be simplified and this will be done in the next section. 2,4 The Pitman Asymptotic Relative Efficiency of V n Relative to Hotelling’s T 2 In this section we will use Pitman relative efficiencies to make comparisons 9 . between V n and Hotelling’s T in the repeated measurement design settings. Because these statistics are all affine-invariant we may, without loss of generality, make the simplifying assumption that the transformed sample variance-covariance matrix is the identity. We will apply the asymptotic results under the contiguous alternatives in Theorem 2.3. 10. The Pitman approach to asymptotic relative efficiency compares two test sequences (Snj) and {T m .} as the sequence of alternatives H, : 0 = 0 it approaches the null, which we are taken to be H 0 : 0 = 0. The subscripts nj and mj are the sample sizes for tests S n . and T m j, respectively. Let (3{S n ., 0j } and p{T m ., 0 j } denote the powers of the tests based on

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25 S n . and T m ., respectively, when 0 = 0j. Assume that nj and mj are such that the two sequences of tests have the same limiting significance level a and ac.lim (3{S n , 0j} = .lim (3{T m , 0j]00 1 Then the Pitman asymptotic relative efficiency (ARE) of {S n .} relative to {T m .} (or simply of S relative to T) is ARE(S, T) = .lim , 1— >00 n j provided the limit exists and is the same for all such sequences {nj} and {nij}, and independent of the {0,} sequence. (See, e.g., Randles and Wolfe [1979], p. 144.) Hannan (1956) shows that if, under the sequence of alternatives Hj, the test sequences { S n . } and [T m .] are asymptotically noncentral Chi-square with the same 2 2 degrees of freedom and noncentrality parameters, 8 S and 5 T , respectively, then ARE(S, T) = It’s well known that under the sequence of contiguous alternatives described in the last section, and taking AIA T = Ip-i, the asymptotic distribution of Hotelling’s T 2 , where T 2 = n Z T (£) _1 Z with Z = n' 1 £ Zj and £= (n-1)’ 1 £ (Zj-Z)(Zj-Z) T , is noncentral i=l i=l chi-square with p-1 degrees of freedom and noncentrality parameter (2.4.1) (See, e.g., Puri and Sen [1971], p. 173.) To derive ARE(V n , T 2 ) we use the following lemma.

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26 Lemma 2.4.2 Let Yj, Y n be i.i.d. with density function given in expression (2.3.1), and R 2 = Z T Z. Taking AIA T = I p _i, then, under Hq : 0 = 0, the density function of R 2 is of the form K tt(P'1)/2 f R 2 (r) =— f r^D^gCr), r > 0, (2.4.3) K r[(p-l)/2] where oo g(r) = j exp { -[(r+s 2 )/Q)] v } ds, (2.4.4) -OO and K p and Co are defined in (2.3.2). Proof of Lemma 2.4.2 Taking A E A T = I p -i and under H 0 : 0 = 0, we have Z is Ep.i(Q, Ip_i) with density function of the form %(z) = K p g(z T z) = K p j exp { [ (z T z+s 2 )/C Q ] v } ds, z e R p_1 . -OO Thus, R 2 = Z T Z has density of the form given in expression (2.4.3). (See, e.g., Muirhead [1982], p. 37.) Theorem 2.4,5 Assume Yj, ..., Y n are i.i.d. from a density given by (2.3.1). In the repeated measurement settings, taking A E A T = Ip.!, if 4v + p > 3, then the Pitman asymptotic relative efficiency of V n relative to Hotelling’s T 2 is ARE(V n , T 2 ) = 4r 4 (p/2)r 2 [(pi)/2v]r[( P +2)/2v] p(p-i)r 4 [(p-i)/2]r 3 ( P /2v) (2.4.6) Proof of Theorem 2.4.5 If 4v + p > 3, taking A E A. T = I p .! and using expression (2.3.11), we have, under the sequence of contiguous alternatives, the test V n is

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27 asymptotically noncentral chi-square with p-1 degrees of freedom and noncentrality parameter Sy , where _±_ p-l Rg'(R 2 ) g(R 2 ) It follows that 4 , ARFfV T^'l 4 If r R g'( R )l) ARE(Vn ’ T) ‘ 5 2 2 pll Vg(R2) (2.4.7) Under Hq, we Find, using expressions (2.4.3) and (2.4.4), = E r R-gW) i E L g(R 2 ) -I VrV(r 2 ) g(R 2 ) [ ] °° V r-g'(r) J— gW f R 2()dr VT-g'(r) K-jifP-')/ 2 i SCO K ttCp-D / 2 £2 r ( P' 1)/2 ‘ 1 g(r)dr r((p-l)/2) r«p-l)/2) )fr p/2 ' 1 g'(r)dr. 0 It can be verified that g'(r) = ( J exp{-[(r+s 2 )/C 0 ] v }ds)' -oo !° r
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28 (See, e.g., Trench [1978], Theorem 5.2 and 5.6 on p. 581 and p. 586, respectively.) Letting 1 E_ p r[(p-i)/2] ’ (2.4.8) we have R-g'(R 2 ) 1 g(R 2 ) J = C p J rP^ 1 | [ 0 -OO 1 °° r 0(exp(-[(r+s 2 )/C o ] v }) 3r ] ds dr OO _ OO = Cp j rP/ 2 ’ 1 j exp{-[(r+s 2 )/C 0 ] v }(-v)((r+ S 2 )/C 0 ) v ' 1 C'Jds dr 0 -OO ( _7 V r' \ oooo — VS S J rP/21 ex P{-[( r+s2 )/C 0 ] v }(r+s 2 ) v ' 1 ds dr. C 0 0 0 Taking r = r, we can write ,2 E [ ' R ' g ( 2 R } J j t p ' J exp { [(t 2 +s 2 )/C 0 ] v } (t 2 +s 2 ) v_ 1 ds dt. (2.4.9) g(R ) Cq oo Letting s = VCq x 1/v sin(0) and t = ^[CQ x 1/v cos (0), we find that 3 s /00 = ^[CQ x 1 /v cos( 0 ), 0s/3x = v _1 VCo x^ 1 ' v )/ v s i n ( 0 ) j 0t/00 = -^Cq x 1/v sin(0), 0t/9x = v' ! VC o x^ 1 " v ^ /v cos(0), and s 2 +t 2 = C 0 x 2/v .

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29 The jacobian of the transformation is v' 1 C o x^ 2 ' v ^ /V cos 2 (0) + v' 1 Cox( 2 ‘ v ^ v sin 2 (0) = v' 1 C 0 x^ 2 ' v ^ /v . Thus, we can write (2.4.9) as r R'g^R 2 ) E L g(R 2 ) -I = ( 7^)j j (VCo xl/V cos(0)) p ' 1 (C o x 2/v ) v ' 1 exp(-x 2 )v' 1 C o x (2 ' v)/v d0dx C 0 0 0 = (-4CpC ( 0 P ' 1)/2 )•( j x (p+v ' 1)/v exp(-x 2 )dx) ( ] / cos p * 1 (0)d0). (2.4.10) 0 o The constant term in the product above is -4CpC^ 1)/2 = -4 K^P -^ 2 r(( P -i)/ 2 ) p(p-l)/2 U 0 (using expression (2.4.8)) 1 vHp/2) r( P /2v)(7tc 0 ) p/2 jcG P^_ r[( P -i)/ 2 ] r (p-D/2 (using expression (2.3.2)) 1 vr(p/2) _1 _ r(p/2v)T[(p-i)/2] V K V Co The first integral in expression (2.4.10) is ~( P+ v4)/v exp( _ x 2 )dx 0 = ~ f y (p " 1)/2v exp(-y)dy (using y = x 2 ) 2 0

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30 — j y^ p+2v ' 1 ^ 2v " 1 exp(-y)dy 2 0 X r( Pl 2 Yd ) = J— r( Eil 2 2v 2 2v _l_(2d) r (2d) = Ml r (Pd 2 2v 2v 4v 2v The second integral in (2.4. 10) is k /2 , jt/2. , ^ n-f-) f cosP'H^Od© = f sinP'^ejdO = — (j J 2 r<-^-) (See, e.g., Beyer [1987], p. 289.) Thus, the expected value in expression (2.4.10) is r Rg '(R 2 ) ] E h^r J = r.4 vr

1 r(p/2v)r((p-l)/2) V n V C 0 4v 2v J L 2T(e±L ) J -(p1 ) r 2 ef-) r(-^-> (2.4.11) Therefore, using expressions (2.4.7) and (2.4.11), we have ARE(V n , T 2 ) -[ n-1 L (P-D r 2 ef) r(-^-) P ‘ 1 2r(^)re| 1 -)re E2i )VQ ] [ ( P -i)r 4 ef)r 2 (-ti-) -. r n-^ 2 -) TH-] • [ —~T~ ] • (using (2.3.2)) r 2 ^ 2 ^)^^) pre£) 2m Noting that

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31 _P±^ _ rr^i rev-) = ne^-)+i] = e — )re^ -), we have ARE(V n ,T 2 ) (P-D r 4 ffo r 2 (-^-> n-ffi pr 3 (^-)r 4 e^)(-ti-) 2 4 r 4 (-f-> r 2 (-E^-) p(p-i) This ARE is evaluted in Table 2. 1 for selected values of v and p satisfying the condition 4v + p > 3. Note that when p > 3, 4v + p > 3 for all v > 0, and when p = 2, 4v + p > 3 for all v > .25. When the underlying population is multivariate normal or close to normal (v = 1.0, .75, respectively), Hotelling’s T 2 performs better, yet V n appears to be quite competitive and the efficiencies increase as p increases. For light-tailed distributions (v = 2, 3, 4, and 5) the sign test is not as effective as T 2 , but the ARE’s increase with p. For heavy-tailed distributions (v = .50, .25, .20, .15, and .10) V n is more effective than T . Here the efficiencies decrease with p. In fact, it can be verified, using Stirling’s formula for approximation of m! and the result lim (1+A.x) 1/,X = e^ for constant X, x— that for fixed v, ARE — > 1 as p — » For fixed p, it can be shown that as v — > 0, the ARE -> °° and as v-> oo, the ARE -» {4p 2 r 4 (p/2)}/{(p+2)(p-l) 3 r 4 ((p-l)/2)}, the latter is evaluated for p = 2 to 12 and is displayed in Table 2.1 under the column v = oo.

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32 n CN T-— < CN * CN CO in >n o CN oo CO CO cn o NO OO OO r-H CO CN oo NO I/O NO or-H *“ H On n cn CO CN CN rH — -* On f" OO _ r~ m in ON CO >n CO o r-H rON in CO CO NO On r-H CO o t"nCN r-H o ON OO • CN CN CN T—i r-H rH r-H o o o rH r-H — 1 — — 1 rH o ON O NO CN oo CN On CN rH in r-H H O On in 00 r-H n f" 00 00 ON ON On On rON ON ON ON ON On ON On On On o o d o o d d d O O O NO O’ OO NO nr CO NO CO NO NO >n OO cn in o r-H On NO wo cn oo noo o CN CO cn in wo r— H NO t-" OO oo On On ON On ON On On o d o o d d o o o o O 00 CO cn r~~ OO NO OO NO On o NO (N n NO oo ON ON ri NO rOO 00 oo OO 00 00 OO o d o o o d d o o d o CN CO r— r~ CO ON o NO o On On CN OO cn 00 cin oo m o tjOn oo CO r-~ o CN NO rOO rn >n NO rroo 00 OO 00 00 OO o d O o o d o d o d d OO CN CN CO o r-» r-~ CN 00 rCN 00 o cn NO in O CN CO V— 2 CN OO NO CN NO ON rH cn m NO r^in NO r» roo oo oo OO oo o d d o d d d o d d o m OO ON nr NO OO CO CN NO wo wo o CO ON NO NO NO o On oo V — •> CN >n 1— 1 in OO r-H CN WO NO in >n NO t" f~ 00 OO 00 oo oo d d o o o o o o d o o CO CN _ OO ro NO t" CO On WO wo o r-~ r00 CO
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CHAPTER 3 A MULTIVARIATE SIGNED-RANK TEST BASED ON INTERDIRECTIONS FOR REPEATED-MEASURES DESIGNS 3.1 Definition of the Test Statistic In this section we describe the multivariate signed-rank test, denoted by W n , proposed by Peters and Randles (1990) for the one-sample multivariate location problem. As explained in section 2.1, this test can also be applied to repeated measures designs for detecting treatment effects. Using the same notations as in chapter 2, we let Yj, ..., Y„ be i.i.d. from a p-variate elliptically symmetric distribution and Zj, ..., Z n be the transformed sample defined in (2.1.3). Recalling the result of Theorem 2.2.5, we have that Z t , ..., Z n are i.i.d. from a (p-l)-variate elliptically symmetric distribution. Thus, it is logical to measure the distance of each observation Zj, i = 1, ..., n, from the origin in terms of elliptical contours and to use the ranks of these distances along with the observations’ directions in forming a test statistic. We now describe such a signed-rank statistic based on V n which includes the univariate signed-rank statistic as a special case. Specifically, let us form estimated Mahalanobis distances via D i = Z T £' 1 Z i ,i=l n, (3.1.1) where 33

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34 1 n j is a consistent estimator of the null hypothesis variance-covariance matrix of Zj, provided it exists, with H 0 defined in (2.1.4). Let Qj = Rank (Dj) among £ji> .... D n , i = 1, n. We now weight the (i, k)th term in the sum V n , defined in (2.1.5), by QjQk, and consider the statistic n i=lk=l Icos(7tp., ) — Qi Qk ik y n n (3.1.2) where P ik is defined in (2.1.6). We reject Hq in favor of H a for large values of the statistic W n . /\ Since the P ik , i, k = 1, ..., n, are invariant with respect to a nonsingular linear transformation (as shown by Randles [1989]) as are the Di, i = 1, .... n, it is clear that W n is likewise affine-invariant. When p = 2, the test based on W n is the two-sided univariate Wilcoxon signed-rank test. For p > 2, W n does not have a small-sample distribution-free property, but its large-sample null distribution is convenient, as is shown in the next section. 3.2 Asymptotics In this section we develop the asymptotic distributional properties of nW n under the class of elliptically symmetric distributions. Assume that Y is elliptically symmetric, that is, the density function of Y is of the form fY(y) = K p lir 1/2 h[(y-kL) T L' 1 (y-U)], ye R p , (3.2.1) where j± is the point of symmetry, I is the variance-covariance matrix of Y, provided it exists, and Kp > 0. The null distribution of nW n is established in the following lemma.

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35 Theorem 3.2.2 Assume the observations Yj, Y n are i.i.d. from a p-variate elliptically symmetric distribution with a density function defined in (3.2.1). Then, in the repeated measurement settings, the test nW n , defined in (3.1.2), has a limiting distribution under Hq, defined in (2.1.4). Proof of Theorem 3.2.2 Note that, under H 0 , taking A Z A T = lp, h the (p-l)x(p-l) identity matrix, Z\, ..., Z^ are i.i.d. as Z = (Zj, ..., Zp.^), where Z is from an elliptically symmetric distribution and can be expressed as Z = RU, where R 2 , as before, equals Z T Z and U is distributed uniformly on the (p-l)-dimensional unit-sphere independent of R. (See, e.g., Johnson [1987].) It can be verified that E[(ZjZ k ) 2 ] = E[R 4 U 2 U 2 ] = E[R 4 ]E[U 2 U 2 ] < E[R 4 ] < oo, for all j, k = 1, ..., p-1. Thus, via the Lindeberg-Levy Central Limit Theorem (see, e.g., Serfling [1980], p. 28), each element of Vn(£ Ip^), ^ 1 n -J> /s where £ = X Z\ Z x , is asymptotic normal under Hq. Therefore, we have Vn(z I p .i) = Op(l) under Hq. (See, e.g., Serfling [1980], p. 8.) So, the test nW n has a limiting y 2 . P-1 distribution under Hq. (See Peters and Randles [1990], p. 553.) Next, we derive the Pitman asymptotic relative efficiency of nW n relative to Hotelling’s T . To do this, we first establish the asymptotic distribution of nW n under contiguous alternatives, as described in section 2.3, for a general class of elliptically symmetric distributions. L gmma 3.2.3 Suppose Xj, ..., X n are i.i.d. from an elliptically symmetric distribution with density of the form fx(x) = K m h[(x U ) T (x u)], x e R m ,

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36 satisfying 0 < I#x) = d T 2fx(x) -|2 %(*) J fxOOdx < oo for all de R m -{0}. (3.2.4) Under the sequence of contiguous alternatives for which X has density of the form fxfe-d/Vn), we have d 2 nW n -> x 2 m ( 12 fr, r K(R 2 )Rh'(R 2 ) Vml W h(R 2 ) ]}Vd), (3.2.5) where H 0 implies U = Q, R 2 = X T X , h'(R 2 ) = dh(R 2 )/dR 2 , and K(t) = P Hq (R 2 < t), t > 0. Proof of Lemma 3.2.3 Let i ; r d T 2f x (Xi) L f x (Xi) ]. * '/3S [ 7 = £ Uj,K(R?) -f £ U im K(R?) ] T Vni=i Vn i=l 1 ~r= X V3m UjK(R?) , Vni=i 1 and S„ l T £, where d, ^ e R m -{0}, and U 4 = [Uji, ..., Uj m ] T is distributed uniformly on the m-dimensional unit-sphere independent of R v Here Xj = RjU; and R 2 = xJXj, and we can write

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37 n d T K m h'(x[x i )-2x i T n = tI [ ^i=l K m h(x[xi) 2h'(R?)Rjd T U; = — z[ 1 i=l L h(R?) Thus, under Hq, T L 1 nJ = 7=2 Vn i=l V3m K(R?)2L T Ui 2h'(Rf)Rid T Ui/h(R2) = ^|[wj (“y). where Vi = V3m K(R?)^ T Uj and Wj = 2h'(R|>R i d T U i /h(R?) 2\ D jTi rVi Note that, under Hq, ^ w . ’s are i.i.d. with : [wj = 2 and Var Vj L Wy an a 12 a 12 g 22J where a n = E(Yf) = 3mE[K 2 (Rf)]-Ea T U i u[2J=2 t 1 2 t since E[K 2 (Rf)] = \ and E(U i U?') = i I mi 022 = E(W 2 )=A{e[ h'(R?)-R, m L L h(R?) ]} d*d,

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38 and K(Rf)Rih'(Rf) h(R f) Thus, a ll a 12 L a 12 a 22) under Hq. (See, e.g., Serfling [1980], Theorem B, p. 28.) Applying LeCamÂ’s Theorems on contiguity (see Hajek and Sidak [1967], p. 212-213), under the condition defined in (3.2.4), we have under the sequence of alternatives. Therefore, under the sequence of alternatives. (See Peters and Randles [1990], Result 1 and Theorem 2, on p. 555 and p. 556, respectively.) Theorem 3.2.6 Assume Yj, ..., Y n are i.i.d. from a density given by (2.3.1). In the repeated measurement settings, taking A Z A T = Ip.!, if 4v + p > 3, then the Pitman asymptotic relative efficiency of nW n relative to HotellingÂ’s T 2 is and hence

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39 16v C 2 ARE(„W n , T 2 ) = ^ ( — ) • c o { OOOOOO r ~ j j j f sP-^P1 (r 2 +t 2 ) v_1 exp{-[(s 2 +u 2 )/C 0 ] v }exp{-[(r 2 +t 2 )/C 0 ] v }ds du dt dr } , 0 0 0 0 (3.2.7) where Cp is defined in (2.4.8) and C 0 , K p are as defined in (2.3.2). Proof of Theorem 3.2,6 Note that, under H 0 , taking A I A T = L.J, Z h .... Z n are i.i.d. with density of the form oo f 2 (z) = K p g(z T z) = Kp j exp { -[(z T z+s 2 )/C 0 ] v } ds, z e RP' 1 . (3.2.8) -OO (See Lemma 2.3.3.) Under the sequence of alternatives, described in section 2.3, Zb •••> Z n are i.i.d. with density %(z c/Vn ), where c e RP' 1 -^}. We have shown that 0 < Ic(%) < 00 if 4v + p > 3 (see Appendix A), thus, under the sequence of alternatives, it follows from Lemma 3.2.3 that d 2 nW n “> (nf v r K(R 2 )R g '(R 2 ) Si1h 0 L g( R 2 ) where g'(R 2 ) = dg(R 2 )/dR 2 and R 2 = Z T Z. Therefore, ARE(nW n , T 2 ) = { E„ [— ^ r2)R§/(r2 ) 1 } 2 P1 H 0 g(R 2 ) JJ Recall Lemma 2.4.2 that R 2 = Z T Z has density of the form (3.2.9)

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40 K te^P1 )/ 2 f R 2 (r) =— j* r > 0. K F[(p-l)/2] Thus, we have „ r K(R 2 )Rg'(R 2 ) 1 W g(R 2 ) J J° [ K(r)Vr g'(r) j f R 2 (r) dr r (p-l)/2-lg( r ) ^ = Cp j°° KWg'WrP/ 2 1 dr. 0 Note that g'(r) = ( j (r+t 2 ) v_1 exp{-[(r+t 2 )/Co] v } dt. c o 0 It follows that [ K(R 2 )Rg / (R 2 ) j ( -2v-C \ 0000 j j K(r)-rP /2 1 (r+t 2 ) v 1 exp{-[(r+t 2 )/C 0 ] V } dt dr C 0 0 0 ( _2v • C \ 0000 r J / [j f R 2 (s) ds ] r p/2 ' 1 (r+t 2 ) v ' 1 exp{-[(r+t 2 )/C 0 ] <-0 0 0 0 (3.2.10) } dt dr. Using the expressions (3.2.10) and (3.2.8) for f R 2 and g, respectively, in the above expression yields

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41 E r K(R 2 )Rg / (R 2 ) 1 H ° L g(R 2 ) J / 4vC p\ -oooor , = ( — ) / J jf s (p 1)/2A r p/2 ' 1 (r+t 2 ) v ' 1 exp { [(s+u 2 )/C 0 ] v } • C 0 0 0 0 0 exp{-[(r+t 2 )/Co] v }ds du dt dr , > 6vC L = ( — ) j j j j s p 2 r p -‘ (r 2 +t 2 ) v "' exp{-((s 2 +u 2 )/C 0 ] v )c 0 oooo exp{-[(r 2 +t 2 )/Co] V } ds du dt dr. Substituting the above expression into (3.2.9) yields Theorem 3.2.6. 3.3 Numerical Evaluation of ARE (nW n . T 2 ) In this section we describe the numerical evaluation of the ARE expression given in Theorem 3.2.6. All calculations were performed with an IBM computer running on a VM/CMS operating system using fortran 77. Simpson’s rule and the IMSL subroutine DMLIN were used to integrate single integrals and 3-dimensional integrals, respectively. Now we describe how the calculations were performed as well as the error associated with them. Letting * -U * -t i * -r u = e , t = e , and r = e , we can write VARE(nW n ,T 2 )

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42 / 12 1 r p / P-1 ' 1 c v ^0 ){ /// j” p 2 (-Cnr*) p ' 1 (r*t*u*), [(/rt’) 2 +(Cnr*) 2 i v -‘0 0 0 0 exp{-[(s 2 +(rmi*) 2 )/C 0 ] v }exp{-[((rnr*) 2 +(^it*) 2 )/Co] v }ds du*dt *dr* } s j j f [ | 4C p s p 'V 1 exp{-[(s 2 +(rnu) 2 )/C 0 ] v } ds ]• 0 0 0 0 [V^^ ( -^ )P ' 1[( ^ )2+( ^ )2]V ‘ 1(t r )' 1 exp{-[((M 2 +(/'nt) 2 )/C 0 ] v }]du dt dr C 0 1 1 1 -(nr j j J [ j k(u, s) dsj h(t, r) du dt dr, 0 0 0 0 (3.3.1) where k(u, s) = 4C p s p ‘ 2 u' 1 exp{-[(s 2 +(/nu) 2 )/C 0 ] v } , h(t ’ r) = VS (-/«r) p ' 1 [(/nt) 2 +(/nr) 2 ] v ’ 1 (t r)1 exp{-[((r«r) 2 +(rnt) 2 )/C 0 ] v }, 4vC p-i r v and (n is the natural logarithm. Note that, following standard arguments and using expressions (2.4.7) and (2.4.9), 1 1 ss 0 0 / /h(t, r) dt dr = V 3ARE(V n , T 2 ) , (3.3.2) where ARE(V n , T ) is given in Theorem 2.4.5. Using Simpson’s rule with an absolute error 3ARE(V n , T 2 ), we can approximate the inner single integral in expression (3.3.1) by g(u, r) satisfying -(nr J k(u, s) ds g(u, r) I < ej/^J 3ARE(V n , T 2 ). (3.3.3)

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43 Hence, using expressions (3.3.2) and (3.3.3), we can write expression (3.3.1) as VARE(nW n , T 2 ) J j f [ g(u> r) ± ei /V 3ARE(V n , T 2 ) ] h(t, r) du dt dr 0 0 0 = jji g(u, r)h(t, r) du dt dr ± [ £l /V 3ARE(V n , T 2 ) ] j 1 /h(t, r) dt dr 0 0 0 0 0 1 1 1 = j j j g(u, r)h(t, r) du dt dr ± e h (3.3.4) 0 0 0 Now, using the IMSL subroutine DMLIN with an absolute error £ 2 , we can approximate the above 3-dimensional integral by VAL satisfying I 111 1 I J j | g(u, r)-h(t, r) du dt dr VAL I < e 2 . (3.3.5) 0 0 0 Combining expressions (3.3.4) and (3.3.5), we have V ARE(nW n , T 2 ) = VAL ± (e^), where VAL satisfies expression (3.3.5). Thus, we approximate ARE(nW n , T 2 ) by (VAL) 2 with the maximum error ERREST = 2VAL(E]+£2) "h (£i+£2)^ (3.3.6) In Table 3.1 we present the asymptotic relative efficiencies for nW n with respect to 2 T for selected values of v and p satisfying the condition 4v+p > 3. We also include ARE(V n , T ), in parentheses, for easy comparison. When the underlying population is multivariate normal (v = 1.0), Hotelling’s T 2 performs well. Both the sign test and signed-rank test appear to be quite competitive, with the signed-rank test slightly better than the sign test provided the dimension is not too large. For light-tailed distributions

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44 (v = 2.0 and 5.0) and p > 2, the signed-rank test W n has greater power compared to both Hotelling’s T and V n . For heavy-tailed distributions (v = .50 and .10), V n is clearly most powerful, although, W n still performs well relative to Hotelling’s T 2 , provided the dimension is not too large. In Table 3.2 we display the values of ERREST, defined in expression (3.3.6), which bound the error in the estimates of ARE(nW n , T 2 ). Assuming Ej = £2 = £, where El and £2 are defined in (3.3.3) and (3.3.5), respectively, we use the foiling e values : E = .001 when v = 1.0, £ = .001 when v 2.0 and p =2 to 6, £ = .005 when v = 2.0 and p =7 or 8, £ = .01 when v = 5.0 and p = 2 to 6, £ = .02 when v = 5.0 and p = 7 or 8, £ = .001 when v = .50, £ = .10 when v = .10 and p = 2, and £ = .015 when v = .10 and p = 3 to 8.

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Table 3.1 ARE(nW n , T^), with ARE(V n , T^) in parentheses. 45 o > o cs o in Dm * — \ — \ v n-s On r-H o oo NO OO oo r~ O CO CO o CO O CN co T-H rCN NO NO ^r r-H r~ r-H poo pS o o t-H* >n n r-H p m CO o t-H pOn in 2 t-H CO o rCN r-H CN CN CN t-H t-H t-H t-H o N m oo rNO CO PS c 13 On On On On On ON ON ‘S3 > * 3 ‘G „ O — \ /^N /— N x— N /— s m « fi S' r-H OO CO CO T— H D OO NO ON o NO CN CN VI ^ p~ CO r-H po cn in NO POO OO OO (N w 'w*' T3 H c ON in CO CO CN a c rCN cn NO oo ON ON CN >* rr-H in NO NO in >n oo © o o o o o II w r-H r-H r-H ^H T-H cx c < /—N — N — s /•— N >n OO ON NO oo CO o CO ON NO NO NO o CN rm T-H >n oo -V 3 oP W tin NO r-; oo o s -^ 'w' N — 'w' C£ ^ ^ c -tf ON oo NO in CO o o CN p~ s ^ NO 00 OO CN m in T3 c OO q r-H CN CN CN (S U w I-I rrl t-H t-H t-H T-H r-H t-H p rd o q V < CN cn -sr in NO P~ oo ‘o' b o ^ U *

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Table 3.2 Error Estimate ERREST of ARE(nW n , T 2 ) 46 oo r>OO to n r~ CN O oo rCN i-H —i o o o o m o
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CHAPTER 4 MONTE CARLO STUDY In this chapter we display results from a Monte Carol study when the dimension is p = 4, the sample size is n = 20, and the significance level is a = .05. In addition to the affine-invariant statistics T , V n , and nW n , we examined two other nonaffine-invariant statistics. The ANOVA F test is included along with the rank transformation test RT, which were both introduced in chapter 1. These five test statistics were compared under five different distributions. They were quadrivariate normal distribution, elliptically symmetric distributions with density of the form given in (2.3.1), Pearson Type II and Type VII (see Johnson [1987], p.lll and p. 1 17-1 18, respectively), and the mixtures of quadrivariate normal distributions. (A quadrivariate normal mixture is obtained by selecting randomly one of two quadrivariate normal distributions. Each of the observations is sampled with probability p from the first distribution and with probability 1 -p from the second distribution. See Johnson [1987], p.56-57.) These distributions were located at Ifa = (m 8 , m5, m 8 , 0), m = 0, 1, 2, 3, for the original sample Yj, ..., Y n , on which ANOVA F test and rank transformation test RT were applied, and they were located at 63 = (m 8 , m 8 , m 8 ), m = 0, 1, 2, 3, for the transformed sample Z lf ..., Z n , on which the tests T , V n , and nW n were performed. The value of 8 was adjusted for different distributions to examine somewhat silimar points on the power curves. Since the performances of tests ANOVA F and RT depend on the variance-covariance structure of the distribution, for each of the above first four distributions, we considered two types of variance-covariance matrices, one with I 4 , the identity matrix, and the other one with a non H-type structure. For the mixtures of normal distributions, we consider one mixture with a non H-type structure, and three other mixtures with H-type structures. In each Monte Carlo simulation, 47

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48 the proportion of times out of 1000 in which each test statistic exceeded the upper a-percentile of its null distribution is reported. The asymptotic null distribution is used to determine the critical value for tests V n and nW n . For tests ANOVA F and rank transformation RT, the null distribution Fj^y is used to determine the critical value. While 2 57 for Hotelling’s T , we use the null distribution — F3 , 17 (namely, a multiple of F3 ,j 7 ) to determine the critical value. All calculations and random variables generations were performed with an IBM computer running on a VM/CMS operating system using fortran 77. Several IMSL subroutines, to be described later in this chapter, were used. In Tables 4.1 and 4.2, the results from the Monte Carol studies for quadrivariate normal distributions with Z = I 4 and Z = E E T , respectively, are presented. Note that we used E = r 1 3 0 -2 1 4 1-1 1 L -2 4 -1 2 1 -1 0 ’ 3 J (4.1.1) producing r 14 -1 -1 22 -2 1 . 16 1 -2 16 -1 1 1 3 -7 -7 30 J (4.1.2) which is non H-type. The IMSL subroutine GGNML was used to generate N(0, 1) variables. As indicated, 1000 samples of size 20 and significance level .05 were used. When Z = I4, as we may expect, ANOVA F and rank transformation test RT have better power than Hotelling’s T 2 . And they all perform better than V n and nW n (see Table 4.1). However, when Z = E E T , Hotelling’s T 2 has good power, followed by V n and nW n , both performing better than F and RT (see Table 4.2). This illustrates the strong dependence of the performances of tests F and RT on the variance-covariance structure of the distribution. For both variance-covariance structures, ANOVA F performs better than RT, and nW n

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49 performs slightly better than V n , which agrees with the result found in Table 3.1 for v =1.0 and p = 4. In Tables 4.3 and 4.4, we display the Monte Carlo results for five members of the class of elliptically symmetric distributions with density of the form given in (2.3.1) with Z=l4 andZ=EE , respectively. Heavy-tailed distributions, v = .10 and .50, the quadrivariate normal distribution, v = 1.0, (repeated in this table to permit comparisons), and light-tailed distributions, v = 50.0 and 100.0, were included. Taking y = 0 and Z = I4 in (2.3.1), it follows from standard arguments that P(R 2 < w) = P(C 0 G 1/v < w), w > 0, where R 2 = Y T Y, Cq is defined in (2.3.2), and G has the distribution of Gamma(l, p/2v). Thus Y can be generated via Y = R U , where R is independent of U having the distribution of a/CqG 1/2v , and U is uniformly distributed on the unit-sphere S p (see Johnson [1987], p. 1 10). Here, the IMSL subroutines GGAMR and GGSPH were used to generate Gamma(l, r) and the variables that are uniformly distributed on the unitsphere S p , respectively. Although it is difficult to compare powers for statistics with different rejection proportions under the null hypothesis, it appears that the Monte Carol results tend to agree with the asymptotic results of the previous two chapters. For light-tailed distributions (v = 50.0 and 100.0), ANOVA F and RT have great power if Z = I4, while HotellingÂ’s T 2 and nW n have the greatest power when Z = EE T . For the heaviest-tailed distribution (v = .10) examined, V n performs best, followed in order by RT and HotellingÂ’s T 2 when Z = I4, and by HotellingÂ’s T 2 and RT when Z = EE T . Generally speaking, for light-tailed distributions (v = 50.0 and 100.0), the signed-rank test nW n performs better than the sign test V n , and ANOVA F performs better than RT; however, for heavy-tailed distributions (v = .10 and .50), the interdirection sign test V n performs better than nW n , and RT works better than F. The superiority of tests ANOVA F and nW n (RT and V n ) for very light-tailed (heavy-tailed) distributions is shown here. Except for the heaviest-tailed distribution (v = .10), ANOVA F

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50 and RT both perform better than V n and nW n when Z = I 4 , however, V n and nW n both perform better than ANOVA F and RT when Z = EE . Also, we notice that RT performs better than Hotelling’s T 2 only when Z = I 4 . Next, we consider the light-tailed Pearson Type II distributions, a special case of elliptically symmetric distributions, with the density function f Y (x) = r(m+3) J Zr 1/2 (i-(y-u) T r 1 (y-n)} m , (4.1.3) — r(m+l)jr having support ^ 1 and shape parameter m > -1. The Monte Carlo results for this family of distributions with Z = I 4 and Z = E E T are presented in Tables 4.5 and 4.6, respectively. Taking \x = 0 and Z = 1^ in (4.1.3), it can be shown that R 2 = Y T Y has the distribution of Beta(2, m+1). Thus, the variable generation is similar to that in the last case considered. (See, Johnson [1987], p. 1 16.) Here, the IMSL subroutine GGBTR was used to generate beta variables. An examination of Tables 4.5 and 4.6 indicates that ANOVA F and RT have great power if Z = I 4 , while Hotelling’s T 2 and nW n have the greatest power when Z = E E . The signed-rank test nW n performs better than the sign test V n , and ANOVA F performs better than RT. Furthermore, ANOVA F and RT both perform better than V n and nW n when Z = I 4 , while V n and nW n both perform better than ANOVA F and RT when Z = E E T . These results agree with the findings in Tables 4.3 and 4.4 for the light-tailed elliptically symmetric distributions. For the heavy-tailed Pearson Type VII distributions, the Monte Carlo results are presented in Tables 4.7 and 4.8 for Z = I 4 and Z = EE^, respectively. The density function has the form of f Y (*) = — ~ 2 'a' 1/2 l 1+ (x-u) T S' 1 (y-u))' m , (4.1.4) — r(m2 ) 7 T where the shape parameter m > 2. Note that, taking ji = 0 and Z = I 4 in (4.1.4), Y can be generated via Y = Z/Vs" (see Johnson [1987], p. 118), where Z is quadrivariate normal

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51 with mean 0 and variance-covariance matrix I 4 , and S is independent of Z having the distribution of % b , b = 2m 4. Note that when m = 2.5, Y has a multivariate cauchy distribution. In Tables 4.7 and 4.8, the sign test V n has the greatest power. Notice that V n performs better than nW n , and RT works better than ANOVA F. Also, V n and nW n both perform better than F and RT when Z = E E T . Note that the same general trends as in the heavy-tailed elliptically symmetric distributions are seen here. Finally in Table 4.9 we examine samples from quadrivariate normal mixtures violating the assumption of elliptical symmetry required for the asymptotic results of the previous two chapters. As we may expect, the sign test V n and signed-rank test nW n appear to be quite robust. For each mixture in Table 4.9, the parameters of the distributions (Ul» U 2 > £i» a °d la) and the mixing probability (p) as well as the parameters of the mixture (ji_and Z ), computed by H = PUl + (1-P)U2> and Z = pZ t + (l-p)Z 2 + p(l-p)(Ui U 2 )(Ui U 2 ) T , (4.1.5) (see, Johnson [1987], p.57), are indicated. In mixture 1, a light-tailed distribution with H-type structure, we see that tests ANOVA F, RT, and Hotelling’s T 2 have great power, and all performing better than V n and nW n . This agrees with that of Table 4.1, for normal distribution with Z = I 4 . An examination of the table for mixtures 2 and 3, heavy-tailed distributions with H-type structure, shows the same general trends. That is, the superiority of tests V n and RT over the signed-rank test nW n and the other competitors are seen. Also, by comparing with the result for elliptically symmetric distribution with v = .10 and Z = I 4 in Table 4.3, and that for Pearson Type VII distribution with m = 3.0 and Z = I 4 in Table 4.7, it illustrates that the performances of Hotelling’s T 2 and ANOVA F are affected by the

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52 asymmetry of the distributions, which as a result makes nW n a better test than HotellingÂ’s 2 T and ANOVA F. Finally, in mixture 4, a heavy-tailed distribution with non H-type structure, the robustness and superiority of tests V n and nW n is easily seen. In summary, for normal to light-tailed symmetric distributions and a variancecovariance structure that is non H-type, HotellingÂ’s T 2 has the greatest power, followed by the signed-rank test nW n . When the structure is H-type and the populations are normal or light-tailed the best is ANOVA F, followed by RT. For the heavy-tailed distributions with H-type variance-covariance structure, the interdirection sign test V n has the greatest power, followed by RT. For heavy-tailed symmetric distributions with non H-type variancecovariance structure, the interdirection sign test V n still has the greatest power, but this time HotellingÂ’s T is second best. For the asymmetric distribution with non H-type variancecovariance structure, the interdirection sign test V n and the signed-rank test nW n are the best. Generally, the tests V n and nW n both perform better than ANOVA F and RT when the variance-covariance structure is non H-type while the contrary is true only for normal to light-tailed distributions with H-type variance-covariance structures. This suggests that V n and nW n are promising procedures for the repeated measures settings. The tests ANOVA F and RT follow the same general patterns as nW n and V n , respectively. That is, the RT (V n ) test performs better than ANOVA F (nW n ) for heavy-tailed distributions, and ANOVA F (nW n ) performs better than RT (V n ) for light-tailed distributions. Also note that test RT works better than HotellingÂ’s T 2 for distributions with H-type variance-covariance structures. Finally, we should point out that the performance of the test based on nW n is disappointing, since the power of nW n is not always better than that of HotellingÂ’s T 2 for light-tailed symmetric distributions (see Tables 4.3 and 4.4). One explaination to this may be because of the higher probability of Type I error of HotellingÂ’s T 2 test.

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53 Table 4.1 Monte Carlo Results for Quadrivariate Normal Distribution with £ = I 4 . Amount of Shift Statistics F RT »p 2 v n nW n .00 .049 .046 .056 .052 .058 .25 .106 .113 .108 .082 .108 .50 .335 .321 .300 .219 .247 .75 .649 .644 .579 .457 .485 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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54 Table 4.2 Monte Carlo Results for Quadrivariate Normal Distribution with Z = E E T . Amount of Shift Statistics F RT T 2 v n nW n .00 .081 .092 .055 .040 .056 .60 .091 .101 .108 .088 .114 1.20 .164 .144 .283 .204 .235 1.80 .279 .248 .540 .430 .453 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5. i

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55 Table 4.3 Monte Carlo Results for Elliptically Symmetric Distributions with Density of the Form Given in (2.3.1) with Z = I 4 . Amount of Shift Statistics F RT T 2 v n nW n v = .10 .00 .026 .048 .016 .032 .040 .17 .079 .226 .097 .276 .124 .34 .248 .636 .367 .669 .288 .51 .500 .872 .626 .894 .432 < 11 Ln 0 .00 .053 .057 .050 .032 .048 .25 .110 .119 .101 .079 .088 .50 .321 .368 .303 .260 .234 .75 .684 .699 .618 .573 .462 v = 1.0 .00 .049 .046 .056 .052 .058 .25 .106 .113 .108 .082 .108 .50 .335 .321 .300 .219 .247 .75 .649 .644 .579 .457 .485

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56 Table 4.3 continued. Amount of Shift Statistics F RT rp2 v n nW n v = 50.0 .00 .046 .049 .059 .032 .044 .25 .100 .085 .092 .057 .087 .50 .316 .279 .247 .166 .261 .75 .656 .594 .559 .361 .568 v = 100.0 .00 .050 .053 .053 .032 .048 .25 .095 .089 .091 .056 .088 .50 .317 .282 .256 .159 .262 .75 .650 .584 .573 .354 .557 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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57 Table 4.4 Monte Carlo Results for Elliptically Symmetric Distributions with Density of the Form Given in (2.3.1) with £ = E E T . Amount of Shift Statistics F RT .p2 v n nW n v = .10 .00 .055 .081 .021 .049 .037 .40 .073 -.128 .094 .277 .127 .80 .124 .277 .325 .657 .268 1.20 .224 .462 .586 .879 .418 v = .50 .00 .079 .081 .060 .049 .063 .60 .094 .104 .110 .100 .107 1.20 .154 .152 .300 .266 .238 1.80 .268 .219 .609 .558 .452 v = 1.0 .00 .081 .092 .055 .040 .056 .60 .091 .101 .108 .088 .114 1.20 .164 .144 .283 .204 .235 1.80 .279 .248 .540 .430 .453

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58 Table 4.4 continued. Amount of Shift Statistics F RT T 2 v n nW n v = 50.0 .00 .078 .079 .060 .049 .048 .80 .110 .105 .139 .098 .124 1.60 .197 .163 .438 .270 .434 2.40 .390 .275 .833 .620 .830 v = 100.0 .00 .073 .080 .065 .049 .055 .80 .106 .109 .138 .099 .123 1.60 .203 .157 .428 .275 .447 2.40 .394 .280 .824 .617 .822 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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59 Table 4.5 Monte Carlo Results for Pearson Type 13 Distribution with Shape Parameter m, and Density of the Form Given in (4.1.3) with £ = 14. Amount of Shift Statistics F RT rji2 v n nW n m = 1.0 .00 .045 .045 .050 .032 .039 .10 .115 .106 .108 .064 .092 .20 .402 .345 .340 .221 .292 .30 .771 .724 .687 .493 .649 m = 2.0 .00 .052 .045 .051 .032 .053 .10 .136 .130 .122 .077 .113 .20 .498 .451 .415 .281 .370 .30 .867 .830 .788 .601 .726 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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60 Table 4.6 Monte Carlo Results for Pearson Type II Distribution with Shape Parameter m, and Density of the Form Given in (4.1.3) with £ = E E T . Amount of Shift Statistics F RT rji2 v„ nW n m = 1.0 .00 .078 .080 .060 .049 .053 .30 .111 .109 .153 .111 .144 .60 .223 .170 .490 .325 .454 .90 .438 .330 .877 .704 .832 m = 2.0 .00 .072 .075 .059 .049 .054 .30 .118 .113 .172 .125 .160 .60 .275 .195 .587 .425 .522 .90 .540 .403 .930 .812 .885 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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61 Table 4.7 Monte Carlo Results for Pearson Type VII Distribution with Shape Parameter m, and Density of the Form Given in (4.1.4) with Z = I4. Amount of Shift Statistics F RT T 2 v n nW n m = 2.5 .00 .019 .052 .022 .052 .050 .40 .024 .106 .042 .107 .097 .80 .067 .262 .109 .287 .190 1.20 .120 .476 .214 .520 .312 m = 3.0 .00 .035 .046 .030 .052 .058 .50 .179 .324 .226 .295 .220 1.00 .592 .863 .661 .829 .575 1.50 .851 .985 .901 .976 .777 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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62 Table 4.8 Monte Carlo Results for Pearson Type VII Distribution with Shape Parameter m, and Density of the Form Given in (4.1.4) with Z = EE T . Amount of Shift Statistics F RT T 2 v n nW n m = 2.5 .00 .025 .077 .027 .040 .048 1.20 .031 .098 .058 .135 .110 2.40 .054 .173 .149 .376 .232 3.60 .087 .294 .281 .653 .375 m = 3.0 .00 .059 .080 .027 .040 .051 1.10 .093 .137 .182 .251 .199 2.20 .228 .340 .567 .709 .508 3.30 .442 .620 .848 .939 .715 Entries : Proportion of times each test statistic exceeded the upper a-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000. Significance Level : a = 0.5.

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63 Table 4.9 Monte Carlo Results for Quadrivariate Normal Mixtures with Mean y, and VarianceCovariance Matrix Z given in (4.1.5). Amount of Shift Statistics F RT »p2 v n nW„ Mixture 1 P = -5, JA] = [1, 1, 1, 1] T = -y.2, Zi = £2 = 14. r 2 1 1 ll * -112 1.1112. .00 .049 .051 .047 .033 .049 .35 .190 .179 .171 .121 .160 .70 .604 .565 .528 .405 .428 1.05 .929 .915 .896 .800 .770 Mixture 2 P = -9, Ui = U2 = Q, I 2 = 400-14 H = Q, Z = 40.9-14. » .00 .021 .050 .031 .046 .050 .35 .037 .151 .078 .102 .125 .70 .096 .427 .231 .336 .317 1.05 .166 .744 .423 .668 .589

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64 Table 4.9 continued. Amount of Shift Statistics F RT T 2 v n nW n Mixture 3 p = .5, m = [1, 1, 1, 1] T = -ji 2 , = 14, Z 2 = 40014, [201.5 1 1 l-i o I1 201.5 1 i ^ y ’ 1 1 201.5 1 • L 1 1 1 201.5. .00 .035 .050 .024 .033 .038 1.40 .055 .229 .047 .332 .149 2.80 .090 .444 .088 .654 .216 4.20 .172 .543 .175 .769 .212 Mixture 4 P = -9, Ui =U2=Q, hi =EE t ,Z 2 = 100EE t , 14 = 0, E = 10.9-E E t , EE t is defined in (4.1.2). .00 .033 .072 .030 .035 .053 1.60 .076 .144 .219 .292 .295 3.20 .224 .402 .617 .834 .723 4.80 .395 .738 .808 .986 .880 Entries . Proportion of times each test statistic exceeded the upper cc-percentile of its null distribution. Dimension : p = 4. Sample Size : n = 20. Number of Samples : rep = 1000.

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65 Table 4.9 continued. Significance Level : a = 0.5. Distributions : Quadrivarite normal mixture, choosing Nj with probability p and N 2 with probability 1-p, where N;, i =1, 2, is quadrivariate normal with mean jij and variance-covariance matrix Zj. The resulting mixture has mean ji and variance-covariance matrix Z (see (4.1.5)).

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CHAPTER 5 A MULTIVARIATE SIGNED SUM TEST BASED ON INTERDIRECTIONS FOR THE ONE-SAMPLE LOCATION PROBLEM 5A Definition of the Test Statistic We now leave the repeated measures problem, and consider the one-sample multivariate location problem. In this section we propose a multivariate signed sum test based on interdirections, described in section 2.1, for the one-sample multivariate location Here 0 is used without loss of generality, since Hq : £ = fio can be tested by subtracting fio from each observation Xj, and testing whether the differences (Xj fio)’s are located at Q. The procedure is somewhat like applying the multivariate sign test, proposed by Randles (1989) for the onesample location problem, to the sums problem. To do so, we let Xi X n , where X* = (X u X ip ) T , be i.i.d. as X = (X h X p ) , where X is from a p-variate absolutely continuous population with location parameter Q (pxl). We would like to test Ho : fl* = Q versus Ha.£* * 0. (5.1.1) Xs + Xt, 1 < s < t < n. (5.1.2) and rejects Hq for large values of the statistic (5.1.3) where 66

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67 if (i, k)*(i’, k') 0 if(i,k) = (i\ k 1 ). (5.1.4) and Cik,i’k' denotes the number of hyperplanes formed by the origin 0 and p-1 of the other observations Xj (excluding Xi, Xk» Xi’> and Xk') such that Xi + Xk and Xi' + Xk' are on opposite sides of the hyperplane formed. The counts {Qk ik I 1 < i < k < n, 1 < i* < k’ < n}, called interdirections, arc used via 7tp ik to measure the angular distance between Xi + Xk and Xi' + Xk 1 relative to the positions of the other observations. Here p ik lk . is the observed fraction of times that Xi + Xk and Xi' + Xkfall on opposite sides of the hyperplanes formed by Q and other p-1 observations. We now examine some characteristics of the test based on SS. First of all, it can be easily seen that the test SS defined in (5.1.3) is like the multivariate sign test V n applied to the sums Xs + Xt> 1 ^ s < t < n. Secondly, since the p^^., 1 < i < k < n, 1 < i’ < k’ < n, are invariant with respect to the nonsinglar linear transformations, as shown by Randles (1989), it is clear that SS is also affine-invariant. Thirdly, for p > 1, SS does not have a small sample distribution-free property. This is because the joint distribution of the directions of Xi + Xk, 1 ^ i ^ k < n, from Q depends on the distribution of the distances of 2ii, 1 < i < n, from the origin. However, its large sample null distribution is convenient, as is shown in the next section. We end this section by proving a Lemma which shows that for p = 1, the test based on SS is the two-sided univariate Wilcoxon signed-rank test. Hence for p = 1 it does have a small sample distribution-free property, as well as a convenient large sample null distribution. Lemma 5.1.5 When p = 1, the test based on SS is the two-sided univariate Wilcoxon signed-rank test.

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68 Proof of Lemma 5.1.5 Note that when p = 1, 1 if (Xj + X k )(X r + X k .) < 0 Pik,i'k' = { 2 if ( X i + X k)( X i’ + X k’) = 0 » 0 if (Xj + X k )(Xj> + X k .) > 0 and thus -1 if (Xj + X k )(Xj+ X k .) < 0 cosC^p^ j.j,,) = | 0 if (Xj + X k )(Xj+ X k 0 = 0 1 if (Xj + X k )(Xj. + X k .) > 0 Hence, we can write co s(KPjk iilc .) = sign (Xj + X k )-sign (Xj+ X k 0 where sign(x) = -1 if x < 0, 0 if x = 0, and 1 if x > 0. Therefore we have SS = 4 —^2 2 X COS(7C P ik j' k ') — 7772 £ 2 si g n ( x i + x k)-sig n ( x i* + x k') n (n+D j
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69 where W' is equal to the number of negative Walsh averages ((Xj + X^/2 , 1 < i < k < n). Now using the fact that with probability one W + . w = Jfo +1 ) t expression (5.1.6) is equivalent to SS = n(n+l)' {W + W) -l 2 — — r { 2W + 2 n(n+l) 2 2 16 n(n+l)“ {w + + n(n+l ) |2 Au+ n(n+l )^2 n(n+l)(2n+l) ^ 4 > 24 n(n+l)(2n+l) 24 n(n+l) 2 16 (W+-E Hf ,(W+))* 2(2n+1 , VarH„(W + ) 3(n+l) ’ since EHo(W*) = and VarjyvT) (See, e.g., Randles and Wolfe [1979], p.56.) This completes the proof. Note that, when p = 1, — > % 2 under Hq as n —> oo. This follows from the fact that

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70 w + ~ E Hq (W + ) d V ar H 0 (W + ) — > N(0, 1) under Hg (See Randles and Wolfe [1979], p.85), and 2(2n+l) 3(n+l) 4— as n — > oo. 5.2 Some Intermediate Results In this section we present some basic and important results which will be useful in the next section for finding the limiting distribution of SS under Hq. Here we discuss some properties involving the sum observations Xj + X^, 1 < i < k < n, when the original sample Xj, 1 < i < n, is from the family of elliptically symmetric distributions. The first one is about the sum and difference of two i.i.d. spherically symmetric random vectors, a special case of elliptically symmetric distributions. Theorem 5.2.1 Let Xi, X 2 be i.i.d. spherically symmetric random vectors. Then X4+X2 and Xj-X2 are spherically symmetric random vectors. Proof of Theorem 5.2. 1 Let D be any pxp orthogonal matrix. It suffices to prove that D(X 1 ±X 2 ) = X 1 ±X 2 (see, e.g., Muirhead [1982], p. 32), where — is read “has the same distribution as”. Note that we have since Xj, X2 are spherically symmetric. Hence, by the independence of Xj and X 2 , we have

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71 Therefore, see, e.g., Randles and Wolfe (1979) , p. 16, g(X 1 ,X 2 ) = g(DX 1 ,DX2), where g(x, y) = x + y. Hence Xj +X 2 = DX 1 + DX 2 = D(X 1 +X 2 ). The difference is handled similarly. This completes the proof. Next we consider the general case of two i.i.d. elliptically symmetric random vectors. Theorem 5.2.2 Let Xj, X 2 be i.i.d. elliptically symmetric random vectors. Then Xj+X 2 and X r X 2 are elliptically symmetric random vectors. Proof of Theorem 5.2. 1 Since Xj, X 2 are i.i.d. elliptically symmetric random vectors, we can write X j = A X j and X 2 = — —2 ’ * * where Xj, X 2 are i.i.d. spherically symmetric random vectors and A is a pxp nonsingular matrix. Thus Xi±X 2 = AXj±AX*=A(X*±X* ). This completes the proof since Xj± X 2 are spherically symmetric random vectors, as shown in Theorem 5.2.1. Define

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72 (5.2.3) where We examine the property of hQ^, X 2 ) in the next theorem. Theorem 5,2.4 LetX.i,X 2 be i.i.d. spherically symmetric random vectors. Then h(Xi, X 2 ) is uniformly distributed on the p-dimentional unit sphere S p . Propf of Theorem 5-2.4 Since Xi,X 2 are i.i.d. spherically symmetric random vectors, Xj+X 2 is also spherically symmetric, as shown in Theorem 5.2.1. Thus it follows that (See, e.g., Muirhead [1982], p. 38, Theorem 1.5.6.) The proof is complete. Note that under the assumptions of Theorem 5.2.4 E Hoto<*'= 2 = E X,( E X 2 tJl«l. x 2 ) I X,]). Define h*(Xi) = Ex k [h(Xi, X k ) I XJ, 1 < i < k < n. (5.2.5) We are interested in the behavior of h*(Xj), 1 < i < n. Theorem 5,2.6 Let X 1? X 2 be i.i.d. spherically symmetric random vectors. Then ^(Xj) is a spherically symmetric random vector. P rpof of Theorem 5,2.6 Let D be any pxp orthogonal matrix. It suffices to show that

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73 D h*(X,) = h*(Xj). (See, e.g., Muirhead [1982], p. 32.) Note that Dh*^) = DExJhQCLX^IXj X + X = D E X2 [ T | -; + 2 2 n I xj (by expression (5.2.3)) _ rPX 1+ DX 2 , “ b X2 L H X! + X 2 II 1 1 r DX,+DX 2 = E X2 L || p x i + d x 2 II * Xi J (II D II = 1 since D is orthogonal) = Ex 2 [h(D X j , D X 2 ) I Xj (by expression (5.2.3)) = Ex 0 [h(P Xj, X 2 ) I Xj ( D X 2 = X 2 by spherical symmetry) = h*(DXj) (by expression (5.2.7)) d * d = h (Xi) (because D Xj = Xj by spherical symmetry). (5.2.7) This completes the proof. Hence, for i.i.d. spherically symmetric random vectors Xj andX 2 , it follows that * we can express h (Xj) as h*(X!) = Ri Uj, * . * where Uj ~ Uniform(Sp) is independent of Rj, a positive random variable. Defining X 2 = EHotQi^Xj^VQC 1 )], (5.2.8) we get

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74 Var-Cov^h*^)] = since ^ =e Ho kr; uI) t (rI ut)] = % 0 kr 1 ) 2 iIn summary we have the following result : if Xj, .... X n are i.i.d. spherically symmetric random vectors, then h*(Xi), h*(X n ), defined in (5.2.5), are i.i.d. spherically symmetric random vectors with Ej^QfOfc)] = Q and Var-Cov Ho [h*(X 1 )] =^-I p , where x 2 is defined in (5.2.8). Expressing Xj = Rj Uj and h*(Xj) = R* U*, 1 < i < n, we now try to relate R* to Rj and U* to Uj. This is stated in the next theorem. Theorem 5,2-9 Let X j , X 2 be i.i.d. spherically symmetric random vectors with X, = Rilli» i = 1, 2, where H, ~ Uniform(Sp) is independent of Rj. Then we can write h*(Xi) = R*(R 1 )H 1 , (5.2.10) U 1 )(R 1 Uj) 1 ] (since =2) E H„K R i ) 2 ]EHo[ui(IiI) T ] Iw®’ )2]i p where

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75 R1/R2+U21 r 1 /r 2 +u 2 i) 2 +i-u2 1 (5.2.11) is independent of Hi, andH 2 = (U 21 , ..., U 2 p ) T . Proof of Theorem 5.2.9 Let X = R U and h*(X) = R* U*. First we want to show that U* = U. Define Ho = (1,0, ...,0) T and Xq = R HoThen h*(Xo) Kq+X 2 II 2Lo+X 2 II I Xq] (by expression (5.2.5)) = E R2.H 2 { "R+R2U21" R2U22 L R 2 U 2 p j N (r+r 2 u 21 ) 2 +r 2 u 22 2 + + kH = e R2.U 2 { R/R2+U21' U22 L U 2p J N (R/R 2 +U 21 ) 2 +U 2 2 +...+U 2 p 7 R/R2+U21" U22 L u 2c N (R/R 2 +U2i) 2 +1-U^ }

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76 (since £ U ?•= 1) j=l " e R2.U 2 i{ e U 22 ,...,U 2d [ 'R/R2+U21' U 22 r 2 . u 2I ] } u 2o J Noting that, given R 2 and U 21 , the expected value of U 2 j,j = 2, p, is zero and thus we write the above expression as t’CSo) = e R 2 ,U 21 { R/^2 + U21 0 L 0 J A^( R^2+U 2 l) 2 +1-U2 1 } = e R 2 ,U 2 i{ = R (R) Uq, R/R 2 +U 21 (R/R2 + U2l) 2 +1-U21 } LoJ (5.2.12) where R is as defined in (5.2.1 1). Let D be any pxp orthogonal matrix. Now we need to show that h (DX 0 )=R (R) DUq. Note that h*(DXo) = E X2 [h(DXo,X 2 )IJCo] , r D Xp+X 2 -i E X 2 [ II D Xn+Xo II 1

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77 r D_ X n +D X, -| d = E x 2 t II D Xq+D, X 2 II 1 (smCC -2 = 2 ) r 2 . Xn+ D X o -1 = Ex 2 L || Xq+X 2 11 1 ^ — 11 = 1 since — 1 S orth °g° nal ) r Xp + Xo -1 Hoi'll = D h*(Xo) = D R*(R) Ho (follows from (5.2.12)) = R*(R)DUo. (5.2.13) Using expressions (5.2.12) and (5.2.13), we have shown that whenever X = R U , h* (X) = R*(R) U. The indenpendence of R*(R) and U follows from the independence of R and U and the fact that R*(R) depends on R only. This completes the proof. All the above theorems and discussions lead to the following result. Let Xj, .... X n be i.i.d. spherically symmetric random vectors with Xj = Rj Hj, 1 < i < n, where LLj ~ U(Sp) is independent of Rj. Then h*(X.i), ..., h*(Xn) are i-i-d. spherically symmetric random vectors with h (X;) = R (Rj) Uj, 1 < i < n, R (Rj) is independent of Uj, 2 Ej^th*^!)] = 0 and Var-Cov^h*^)] I p> where R*(Rj) and T 2 are defined in (5.2.1 1) and (5.2.8), respectively. 5.3 Asymptotic Null Distribution of SS In this section we establish the null limiting distribution of SS under the class of elliptically symmetric distributions, with density function as defined in (3.2.1). Since the

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78 test based on SS is invariant with respect to nonsingular linear transformations, it suffices to consider X i = R i U i , 1 oo. Proof of Theorem 5.3.3 In proving this result we will need to show E H q[(SS SS* ) 2 ] — ^ Oasn-^oo Note that, using expressions (5.1.3) and (5.3.2), .2 E^KSS SS* ) 2 ] = z ESS E h {[cosfnp^On (n+i) i<&i'
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79 It can be easily shown that if (i, k) = (i', k') or (s, t) = (s', t’), then the term is zero. Also, if any one of the pairs (i, k), (i', k’), (s, t) or (s', t') is disjoint from the integers in the others, then the expected value is zero. The outline of this proof is given in Appendix B. An examination of the terms in (5.3.4) shows that, the number of terms with no disjoint pair is of the order n 6 . This is explained as follows. Suppose pair (i, k) is not disjoint from the integers in the other pairs, then there exists a pair with at least one common integer as in (i, k). Without loss of generality, let us assume this pair is (i, k'). Now the pair (s, t) could be disjoint from pairs (i, k) or (i, k'), but must have at least one integer common with that of (s', t'). (Otherwise pair (s, t) will be disjoint.) Let us call pair (s’, t') as (s, t'). Thus the number of terms with no disjoint pair is v v v v i rc(n+ 1 ) _ 1 ) n _ 6 iik . oc ik ,ik *l • I 7 C p S t,s't' a st , s 't'l } . Note that Pik.i'k' ‘ a ik,i'k'l‘ Itt Pst.s't' ~ CC s t,s't'l — 2-2 = 4 a.e. Furthermore, the n p's are consisent estimators of their respective a's. Thus, by the Lebesgue Dominated Convergence Theorem, see, e.g., Chow and Teicher (1978), p. 99, we have e Hq{ | 71 Pik.i'k' • a ik,i'k'Mrc Pst.s’t' ast.s’t'l } “ > 0 as n -> 00. Hence we have proved

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80 E Ho [(SS SS* ) 2 ] — > 0 as n — > oo. This completes the proof, since convergence in quadratic mean implies convergence in probability. >|c The next theorem proves the limiting distribution of SS under Hq. Theorem 5.3.5 If the observations Xj, 1 < i < n, are as defined in (5.3.1) and Hq is true, then SS* 4 x 2 as n — > oo ; where x 2 is defined (5.2.8). Proof of Theorem 5.3.5 Note that a ik,i'k’ = angle between (Xi+X k ) and (Xj'+X k -) = angle between Xi+X k II Xi+X k II X p+X k ' II Xi'+Xk’ II Hence, we can write cos(a ik>i ' k 0 = ( Xj+X k £j' + X. k X.i'+2Lk since Xi+Xjc II Xi+2Lk II has norm 1. Now SS* in (5.3.2) can be written as SS* = 4 P y v ( — j+X-k \T / Kj'+X k ' N n(n+l) 2 ^ i( < k ' " i+ ~ k " ; ^ " Ki ' +Kk ’ " ^ = 4 P n(n+l) 2 (Z-Ii i
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81 _ I V n X 4 +X.IC T v n ^ 2Li’+X k ' PV n(n+ 1 ) ^ II Xi+Xv II ; V n(n + l) L II X;.+X t . II > ' 2 i^k ~ 1 ~ k 2 i'^k' ~ 1 ~ k = p-R R , v — n— n’ (5.3.6) where R_ = V n n(n+ 1 ) 2 i 00 . Using the fact that —> 1 and — > 0 as n-+ 00 , we see that, under Hq, expression (5.3.7) implies R n Vn U ln —> 0 as n — > 00 . (5.3.8)

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82 Combining expressions (5.3.6) and (5.3.8), now it suffices to find the limiting distribution of Vii Uj n under Hq. Define lll n = \ Xh*(Xi), (5.3.9) 1=1 $ where h is as defined in (5.2.5). It follows from Theorem 3.3.13 (Randles and Wolfe [1979], p. 82) that, under Hq, Vn (U ln U ln ) 0 as n -> oo, and hence expression (5.3.8) implies Rn-^ y;„ Note that, under Hq, VH u; n = VH where h (Xj)'s are i.i.d. with 2 E Ho [h*(X,)] = Q and Varj^Ih'QC,)] =— I p (see section 5.2). Hence by the multivariate central limit theorem we have, under Ho, Vn U* n N p (0,-y-I p ) as n — > oo. (5.3.11) Using expressions (5.3.10) and (5.3.11), it follows from Slutsky's Theorem that, E.n Np(Q,-y-I p ) as n — > oo, p — ^Oasn— »oo. (5.3.10) ( ; • X 2k*(Xi)), i=l

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83 and hence -^•Rn N p (0, I p ) as n — > oo. Therefore, under Hq, using (5.3.6), we have -^=(#En) T (#K„) 4 X *asn 4x" 2x 2x — > oo. This completes the proof. Now we are ready to state the theorem for the limiting distribution of SS under Hq. Theorem 5.3.12 If the observations Xj, 1 < i < n, are as defined in (5.3.1) and Hq is true, then SS A 2 , —3 > X D as n -» oo, 4x z p where x 2 is defined (5.2.8). Proof of Theorem 5.3.12 With Theorems 5.3.3 and 5.3.5 established, the result follows directly from the Slutsky's Theorem. Note that when p = 1, x 2 = 1/3. Thus when p = 1, SS 3 cc A 2 ~^T~ SS-^Xiasn->°°. Finally, to perform the test will require a consistent estimator of t", say The test will then be based on the fact that under Hq,

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84 as n — » oo. 4? For consistent estimator of x , we consider ^ ~ n(n-l)(n-2) ^ ^ cos(7X Pik,ik’)» (5.3.13) i=l k 0 as n — > oo. Note that 't 2 = E Hn [(h*(Xi)) T h*(X 1 )] (by expression (5.2.8)) _ r (X 1 +X 2 ) t (X 1 +X 3 ) = Ef^L II X.i+X .2 II II X!+X 3 ii J (b y ex P ression s (5.2.3) and (5.2.8))

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85 r (Xi+X k ) T (Xi+X k .) . = E H 0 [ II Xj+X k II II X-i+X-ic' II I* 1 1 n> k < k ', k * l, k' * i. = Ej^tgCXp X k , X k 0] , (5.3.15) where fiCXi. X k) X k 0 = (Xi+x k ) T (x i+ x k o II Xi+X k II II Xj+Xk' II satisfies g(Xj, X k , X k >) = g(Xj, X k s Xk). Also we can write n(n-l)(n-2) ^ ^ cos ( a ik,ik’) i=l k• where «*(Xi. Xk, x k .) = l { (X i +X k ) T (X i +X k .) (X k +Xi) T (X k +X k .) 3 l II X i+ X k II II Xi+X k . II + II 2Ck+X, II II Xk+Xk' II

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86 (X k '+X i ) T (X ) ,+X k ) 1 II X k .+Xi II II X k .+X k II J (5.3.16) { |(Xi, x k , 2£k’) + g(Xk, Xi, X k )+ g(X k ., Xi, &) } is symmetric in its arguments. Thus, by Corollary 3.2.5, Randles and Wolfe (1979), p. 71, we have, under Hq, x 2 x 2 — ^ 0 as n — > °°. 5.4 Asymptotic Distribution of SS under Contiguous Alternatives In this section we establish the asymptotic distribution of SS under a sequence of $ alternatives approaching the null hypothesis Hq : 0 = 0 for a specific class of elliptically symmetric distributions. As a first step, let us assume Xi, ..., Xn are i-i-d. as X = (Xj, ..., X p ) , where X is elliptically symmetric with a density function fx of the form fx(i)=K p l£r w exp(-[(x-fi’) T £1 (i-fi')/C 0 ] v },xs RP, (5.4.1) $ where K p , Cq are as defined in (2.3.2), £ is the point of symmetry, and E is the variancecovariance matrix. Since both SS and Hotelling’s T 2 are affine-invariant, we can, without loss of generality, assume that I = I p , the pxp identity matrix. Thus, under H 0 , X], ..., X n are i.i.d. with density of the form fx&) = K p -exp{-[(x T x)/Co] V }. (5.4.2)

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87 Under the sequence of alternatives considered in section 2.3, Xq, X^ ^ i-i-d. with density fx(x-cn' 1/2 ), where fx is given in (5.4.2) and c e R p -{0) is arbitrary, but fixed. Substituting p+1 for p in the proof of Appendix A, we have shown that if 4v + p > 2, then r c T 9fx(x) -i 2 0 < I £ (fx) = } [ f x ^) ] fx(x)dx < o° for all c ^ 0. (5.4.3) Thus, when 4v + p > 2, the previous results in conjunction with LeCam's Theorems on contiguity (see Hajek and Sidak [1967] pages 212-213) can be used to establish the asymptotic distribution of SS under the sequence of contiguous alternatives. This is stated in the next theorem. Theorem 5.4,4 Let Xq, ..., X n be i.i.d. from an elliptically symmetric distribution with density function fx given in (5.4.1) with X = I p , the pxp identity matrix. If 4v + p > 2, then under the sequence of contiguous alternatives for which X has density of the form fxOi-cn' 1 ^ 2 ), defined in (5.4.2), we have SS 4t 2 4v 2 (-^-f E Xp Vpcf 1 H 0 [R 2V-1 R*(R) ]}V £ ), as n — > °o, (5.4.5) O * where x and R (R) are as defined in (5.2.8) and (5.2.1 1), respectively. Proof of Theorem 5.4,4 Define 1 " T £ T 9fx(Xi) ] f X^>) J> n * Ih (Xj), i=l and

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88 s„ = vs i T u; n , where c , X e R p { 0 } , and h* is defined in (5.2.5). Here X; = Rj Uj, where Uj is 2 T distributed uniformly on the p-dimensional unit-sphere independent of Rj, and R ; = Xj X;, thus we can write „ i fi T SO, T &)^ CJVH i=i T = 1 n — 2v i tV'-'a CqVH i=l Thus, under Hq, El— y 2l T h (Xj) — c T R i 2v ' 1 ijj v i — X VSfei 21 t R (Rj)Uj -c T Ri 2v -' Hi C 0 (since h (Xj) = R (Rj) Uj, as proved in Theorem 5.2.9) _J_ " r Ai Vn i= i L B i _ (say), where Aj = 2^ T R*(Ri) Ui and Bj = — c T R i 2v ‘ 1 Uj. r v '“O Note that, under Hq, are i.i.d. with

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89 E r A ii L B iJ = Q and Var a ll °12 a 12 °22-T where an = E(A 2 ) = E(2k T R*(Rj) Ui -2R*(Ri) U; T 2J = 4E{[R*(R i )]} 2 ^ T E(U i U 7 )^ = E{ [R*(Rj)] } 2 -A, t 2-, a 22 = E(B 2 ) = — ^ E(R i 4v ' 2 )-c T E(U i U 7 )c C 0 = -^-E(Ri 4v 2 )-c T c, P C 0 and an = E(AjBj) = E(2X, T R*(Rj) Uj~ Rj 2v *^ Uj 7 c) c o = ^E(R i 2v 1 R*(R i )).2 t T E(U i U 7 )c C 0 = ~~ E[R i 2v ' 1 R*(R i )]2t T c. PC 0

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90 Thus, under Hq, see, e.g., Serfling(1980), Theorem B, p. 28, °12 a 22as n — » oo. where a^, ( 722 , and are as defined above. Applying LeCam’s Theorems on contiguity (see Hajek and Sidak [1967], p. 212-213), if 4v + p > 2, then we have, under the sequence of alternatives, d S n — > N(o 12 , a n ) as n — > oo. That is, under the sequence of alternatives, S n = Vn l T u\ n 4 n(-^ E[R i 2v ' 1 R*(R i )]-^ T £, pC, 4r P l T k) as n — > oo since E{[R*(Ri )]} 2 = T 2 . Therefore, under the sequence of alternatives, •fr, uj n 4 N P (^; E[Rj^ v ''R*(Ri)] c, ^ Ip) as n -> P c 0 and hence ^VSu; n 4 Np ( 7 4 1 ;E[R>1 R*(R i )]. £ .I p ) as n V ptCq — > oo. Recall that R n Vn U 1n *4 0 as n — » oo and SS* = p-R T R 11 — In r — n — n

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91 (see expressions (5.3.8) and (5.3.6), respectively). Using the Slutsky’s Theorem we see that R n and Vn U ln have the same limiting distribution, and thus, under the sequence of alternatives, we have. SS 4x 2 V* t a 4 x = jt-A) 2x ^ 2x ^ “^Xd('T^;{ E hJ r2V 1r *( r )] } Vs), asn “ p V xVf H ° — * oo. Therefore, under the sequence of contiguous alternatives. SS 4t 2 4y 2 (^-{e ^ V 9 „2v l n ^ p N 2 pc n 2v H 0 [r 2v_1 R*(R) ]}Vs), as n — > oo. since SS 4x 2 SS* An — 7 0 as n — > oo 4x 2 (proved in Theorem 5.3.3). This completes the proof. Finally, we establish the Pitman asymptotic relative efficiency of the test based on SS relative to Hotelling’s T 2 in the next theorem. Theorem 5-4.6 Let Xj, ..., X n be i.i.d. from an elliptically symmetric distribution with density function f^ given in (5.4.1) with Z= I p , the pxp identity matrix. If 4v + p > 2,

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92 then the Pitman asymptotic relative efficiency of the test based on SS relative to Hotelling’s T 2 is ARE( ^4 , T 2 ) = 4x z {e Ho [r 2v -‘r*(R)]} 2 , (5.4.7) where x and R (R) are as defined in (5.2.8) and (5.2.11), respectively. Proof of Theorem 5.4,6 Since under the sequence of contiguous alternatives. the result follows by taking the ratio of the noncentrality parameter of the test SS to the In this section I describe the numerical evaluation of the ARE expression given in (RCI) account offered by the Northeast Regional Data Center (NERDC), running on an IBM computer using fortran 77. A subroutine called ADAPT (see Genz and Malik [1980]) was used for numerical integration over an N-dimensional rectangular region. Now we briefly describe how the calculations were performed. First note that, following some standard transformations and arguments, we can express noncentrality parameter of the test T . 5.5 Numerical Evaluation of ARE fSSMx 2 T 2 ) Theorem 5.4.6. All calculations were performed with a Research Computing Initiative K p 7 iPc o +V " 1/2 1 1 j j f(x, y) dxdy, 0 0 (5.5.1)

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93 and x 2 K p 7t p/2 C p/2 v-F(|) v ill ) III g(x, y, z) dxdydz, 0 0 0 where f(x, y) = (-/nx) {-[ny) 1 p-1 p-2v ] 2v )(-[nx) 2v(./ny) 2v (5.5.2) g(x, y, z) = k( [ f ^ ] 2v ) k([ ) L r n y ] 2v )-(-/nx) 2v (.( n y) 2v (.foz) 2v (-/nx) p-2v E^y E^y (-^y) (-/"nz) K s+cos(8) sin p (8)T(^-) 0 (s 2 +2s-cos( 8)+1) 1/2 r(^)7i 1/2 (5.5.3) K p and Cq are as defined in (2.3.2), and In is the natural logarithm. Using expressions (5.5.1) and (5.5.2), we can simplify expression (5.4.7) to [ | j f(x, y)dxdy] 2 ARE(SS/4t 2 , T 2 ) = constant— j-y-j , (5.5.4) j j j g(x, y, z)dxdydz 0 0 0 where 4v 2 n^) constant = — — — — , (5.5.5) p 2 r 2 (^) and f, g are as defined in (5.5.3). Now using subroutine ADAPT with a relative error 1 1 ESTREL, we can approximate j j f(x, y)dxdy by VAL satisfying 0 0 1 1 I j j f(x, y) dxdy VAL I < ERROR, 0 0

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94 where ERROR = VAL x ESTREL. Hence we approximate [ j j f(x, y) dxdy] 2 by (VAL) 2 0 0 with the maximum error ERROR 1 = 2VALERROR + (ERROR) 2 . (5.5.6) 1 1 1 Also, we approximate j j j g(x, y, z) dxdydz by TAU satisfying 0 0 0 1 1 1 I j j j g(x, y, z) dxdydz TAU I < ERROR2, (5.5.7) 0 0 0 where ERROR2 = TAUxESTREL. So we can write expression (5.5.4) as ARE(SS/4t 2 , T 2 ) = constant-^ (VAL) 2 ± ERROR 1 , TAU ± ERROR2 ’ and hence approximate ARE(SS/4t 2 , T 2 ) by constant((VAL) 2 /TAU) with the maximum error ERREST = constant ERROR1TAU + ERROR2(VAL) 2 TAU(TAU ERROR2) (5.5.8) Note that when p = 1, using the expression (5.4.6) on p.166 of Randles and Wolfe (1979), the ARE expression in (5.4.7) can be simplified to ARE(SS/4x 2 , T 2 ) = 12v 2 T(^) 2 1/v r 3 (i) (5.5.9) In Table 5.1 we present the asymptotic relative efficiency of the interdirection signed sum test relative to Hotelling’s T for selected values of v and p. When p = 1, the ARE’s are evaluated from expression (5.5.9). For p = 2 to 5, and thus 4v+p > 2 for all v > 0, the ARE’s are numerically evaluated using expressions (5.5.4) to (5.5.7). When the underlying population is multivariate normal (v = 1.0), Hotelling’s T 2 performs better, yet

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95 the test based on SS appears to be quite competitive. For lighted-tailed distributions (v = 2.0 to 5.0), HotellingÂ’s T 2 still has better power. For heavy-tailed distributions (v = .75, .50, .25, and .20), the interdirection signed sum test is more effective than HotellingÂ’s T . In Table 5.2 we display the values of ERREST, defined in (5.5.8), which bound the error in the estimates of ARE(SS/4x 2 , T 2 ) for p = 2 to 5.

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96

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97 (N in H w c* < H 00 w C* w 2 W o fc W

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APPENDIX A In the context of theorem 2.3.10, we show that if 4v + p > 3, then r c T 3fv(z) -.9 0 < I £ (f z ) = J [ ' J f z(z) d z < 00 for all c * 0. Proof Under Hq : Q = Q, taking A £ A T = I p .i, Z has density function f z(z) = K p f exp { [ ^ q+ S -] V } ds, and thus, following standard arguments, we have 2f z fe) = Kp @ eX p{-[-%i]v} ds . Therefore, yf z ) r c T 3f z (Z) -I, L f z (Z) J rT'y , „2 fz(Z)^ L Co 0 ) Now, it suffices to show that 0
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99 Case 1 : If v < 1, then E ( -p(-^f i ) v ) d ^ 2 } s E ( [-S) ]2 r4r ]2v ' 2[ [ ^ = %M^ 2 ^) 2v ~ 2 } = [^] 2 c 2 0 2v e{ £ t uu t £ }e{r 4v ' 2 } T 7 _lc2 (This step follows from the fact that Z = RIJ, Z T Z = R 2 , and the independence of R and 1J.) [^-] 2c2 0 2 '' e (^A}e{ r4v 2 }. P-1 So now it suffices to show that E[R 4v ' 2 ] < Note that R 2 has density function of the form K n 7t (p ‘ 1)/2 f R 2(D =F[(p-l)/2] . r (p-l)/2-l | exp { -[(r+s 2 )/C 0 J V } ds. Thus e[r 4v ' 2 ] = e[(r 2 ) 2v ' 1 ] 9V ir(P-D/2 oo oo Tkp-D/Z) JJ 1+ (P1)/2* 1 , exp { [( r +s 2 )/Co] v ) ds dr

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100 9 Letting r = t , we have { „ -i ak r 4v21 ( [ ,4v+p-2 exp{[( t 2 + s 2 ) / C 0 ]V)ds d t. r[ /v exp(-x 2 )dx } . i Note that the first integral in the above expression 71/2 J (cos(0)) 4v+p ' 4 d0 < oo. if 4v+p-4 > -1, i.e., if 4v+p > 3. Also, letting y = x“, we have the second integral "x(3v+P-2)/v eX p(-x 2 ) dx = I Ty(4v-2+p)/2v-l exp( . y) dy < 00> i J if (4v-2+p)/2v > 0. Note that 4v-2+p is always positive since we consider v > 0 and p > 2. So we have proved that, if 0 < v < 1 and 4v+p > 3, E { R 4v ' 2 } < oo, (A. 2) and hence I £ (f z ) < oo.

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101 That is if p > 3 and 0 < v < 1 if p = 2 and ^ < v < 1 (A. 3) Case 2 : If v > 1, then we can write the integral in (A.l), provided it exists, as eX p ds 1 7T7 , 2 co Z T Z+s 2 \vi , °°/ Z T Z+s“ v1 r / Z T Z+s 2 \vi j ) } ds+ | ( (^ — V exp{(-g o — ) v ] ds = A + B, where A = 1 Z T Z+s 2 w .i Z T Z+s 2 W1 J (_ Q, )V ' ex P f ( “c 0 > > ds ^ ^ i Z T Z+1 \v-i r f Z T Z+s“ \vi j / • 1 , < J (^ ^ — ) exp { ( ~ ) } ds ( since v > 1) t Z.^Z.+ 1 w— 1 ^ r / Z T Z+s 2 1 j = (— cT _) j expl ~ ( Co ) ids ’ and B °° Z T Z+s 2 NV .i Z T Z+s 2 w / « P {-e^^) v )ds S j s(^ T | +s2 ) v ' 1 exp{( ) v } ds ( since s > 1)

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102 C 0 °°, Z T Z+s 2 w-l , 2s x f / Z T Z+s 2 wi T / ( — cT") < q > ex P*-< c 0 > 1 ds Co r f / Z Z+S i l°°i -[-exp((c — ) }lj ] c o f /Z T z+i w1 — exp{-( — ^ — ) }. 2v Co Note that z T z+s 2 1 y 1 fz(Z) = 2K P J ex P(~ [ ~ Cq a ] )ds > 2K p j exp{[ ~ ] v } ds 1 Z T Z+1 -J V 1 = 2K p exp{[r 7 J1 3 V ). > 2K p jexp{-t^ Li -] v }ds z T z+l C 0 Therefore A < |z T z+l j y-i 1 B .C 0 1 fz(Z) ~ L Cq and r ( r y\ — ’ 2K n f Z© 2v 2K r Thus, by (A.l), we have E {t^i)] 2 [ A+B ] 2 } ^E{[( £ T a 2 [t i ^ tL ) v -.+^] 2 } ^-rE{ £ T iJU T £}E{R 2 [e 4K n 2 1 J L R z +1 C 0 -)V"1+— l 2 | 2v J

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103 (Since Z = RU, Z T Z = R 2 , and R is independent of and U.) So now it suffice to show that e{r 2 [<^) v -‘ + ^] 2 }<~. Let g(R 2 ) . R2 [e ^L)V-, + So ]2 . { R[f ^±L)v-. + So ] } 2. M) 2v M) 2v Note that g is a function of R of order 2v -1. Using the result that “ If X e £p, then X e for all 0 < q < p.”, it suffice to show that e[(R^)^ v '^] < From (A. 2), we see that the result holds if 4v+p > 3. Note that 4v+p > 3 is always true since we are now considering v > 1 and p > 2. Thus, Ij.(f z )<~ if v > 1. (A.4) Combining (A.3) and (A.4), we have proved I £ (fz) < 00 if 4v+p > 3. That is i £ (fz) < ~ if p > 3 and v > 0 if p = 2 and v > ^ (A. 5) ( II ) We now need to show I £ (f z ) > 0. From (A.l), it suffices to show that it / T c T Z 12 r °°/ Z T Z+s 2 w-i r / Z T Z+s 2 \vi j 12 1 a E i L W J 2 [ J ( ~Ca ) ex P{(~^ )} ds ] 2 ) > o. (3 C 0 Co

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104 00 7^74-s^ 7T7 , „2 Since fz(Z) > 0 and h(Z) = j (~ ) v_1 exp{( ^ ) v } ds > 0 V Z, we need to show E{ (c T Z) g(Z) } 2 > 0 V c * 0, where g(Z) = h(Z)/fz(Z). (A.6) To verify (A.6), we note that if 3 c * 0 s.t. E{(c T Z) g(Z)} 2 < 0 ^ {(£ T Z) g(Z)) 2 = 0 a.e. =7 (£ T Z) g(Z) = 0 a.e. => (£ T Z) = 0 a.e. (since g(Z) > 0 V Z) => (£ T LD = 0 a.e. (since Z = RU and R > 0) => Var(c T U) = 0 =* c? Var(Uj) = 0 j=l J (Since c T = (cj, ..., Cp^), U T = (U lt ..., Up.!), and Uj's are uncorrelated.) => Cj = 0 V j = 1, ..., p-1. A contradiction. This completes the proof.

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APPENDIX B Here, we prove in the context of Theorem 5.4.4 that if any one of the pairs (i, k), (i’, k'), (s, t) or (s', t') is disjoint from the integers in the others, then the expected value E^UcosOt Pik,i'k') ' cosCocik.i’k')] * [cos(tc pst.s't’) cos(a st ,s't')] } is zero. Proof Suppose (i, k) is unique (i.e. disjoint), then E{ [cos(7tp ik ,i-kO-cos(a i k > i-k')]-[cos(7tp stjS ' t ')-cos(a sts -f)] } E ( E Xi, Xkft cos(7t Pik.i'k')cos ( a ik,i v)] ’ [cos(7tp st s ' t ')-cos(a st>s ’ t 0] I all Xj except j = i and k) } (B.l) Note that E Xj, xj [ cos (^ik,iv)-cos(a ikti . k .)][cos(7tp st>s ' t ')-cos(a st)S ' t ')]l all Xj except j = i and k} = E Dj, Aj, D k , Ak t [cosCTtpCDjAi, D k A k , X iS X k -))-cos(a(D i A i , D k A k , Xjs Xk’))]-[cos(7tp styS ' t ')-cos(a stiS ' t 0]l all Xj except j = i and k} = E Aj, A k { E Di, D k [ c °s(^p(DjAi. D k A k , X r , Xk’))-cos(a(DjAi, D k A k , X r , X k '))l Ai, Ak]-[cos(jtp st>s t ')-cos(a sts ' t ')]l all Xj except j = i and k} ,(B.2) 105

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106 where Dj = signCX^), Aj = sign(X il )X i . Now conditioning on Aj, Ak and all X, with j * i or k, we have E Di, D k [ C0S (a(DjAi, D k Ak, X r , Xk'))] = { cos(a(Ai, Ak, X iS X k ')) P(Dj = 1, Dj, = 1) + cos(a(Aj, -Ak, X iS 2CkO) P(Di = 1, D k = -1) + cos(a(-Ai, Ak, Xj-, XkO) P(Di = -1, D k = 1) + cos(a(-Ai, -Ak, X r , X k .)) P(Dj = -1, D k = -1) } . (B.3) Note that D, and D k are conditionally i.i.d. with P(D, = 1) = P(Dj = -1) =^. Note also that cos(a(Aj, A k , X iS XkO) = -cos(a(-Ai, A k , X iS XkO), and cos(a(-Ai, -A k , Xi', X k -)) = -cos(a(Ai, -Ak, Xi', Xk'))Thus, the expression in (B.3) equals 0. Similarly, cos(7tp(Aj, Ak, X is Xk-)) = -cos(7tp(-Ai, Ak, Xi', X k 0), and cos(jtp(-Ai, -Ak, Xi-, X k 0) = -cos(7tp(Ai, -A k , Xr, X k 0), since, for example, p(-Ai, Ak, Xi', Xk') = 1-pCAi, Ak, Xi', X k 0. Therefore, following from the same arguments made above, conditioning on Aj, Ak and all Xj with j * i or k, EDj, D k [cos(icp(D i Ai, D k Ak, Xi-, 2Ck ))] = 0. (B.4) Hence, using expressions (B.2) (B.4), expression (B.l) implies that e Hq{ Pik,i'k') ' cos(aik,i , k , )]-[cos(7t Pst.s't') cos(a st , S 't')] } = 0. This completes the proof.

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BIBLIOGRAPHY Agresti, A., and Pendergast, J. (1986). Comparing Mean Ranks for Repeated Measures Data. Communications in Statistics. 15, 1417-1433. Bennett, B. M. (1962). On Multivariate Sign Tests. Journal of the Roval Statistical Society . Ser. B, 24, 159-161. Bennett, B. M. (1964). A Bivariate Signed Rank Test. Journal of the Roval Statistical Society . Ser. B, 26, 457-461. Beyer, W. H. (1987). CRC Standard Mathematical Tables . CRC Press, Inc. Bickel, P. J. (1965). On Some Asymptotic Competitors to HotellingÂ’s T 2 . Annals of Mathematical Statistics . 36, 160-173. Blumen, I. (1958). A New Bivariate Sign Test. Journal of the American Statistical Association . 53, 448-456. Box, G. E. P. (1954). Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, II. Effects of Inequality of Variance and of Correlation Between Errors in the Two-Way Classification. Annals of Mathematical Statistics . 25, 484-498. Brown, B. M., and Hettmansperger, T. P. (1985). Affine Invariant Rank Methods in the Bivariate Location Model. Technical Report #58. The Pennsylvania State University. Chatteijee, S. K. (1966). A Bivariate Sign Test for Location. Annals of Mathematical Statistics . 37, 1771-1782. Chow, Y. S., and Teicher, H. (1978). Probability Theory: Independence. Interchangeability. Martingales . Springer-Verlag, New York Inc. Dietz, E. J. (1982). Bivariate Nonparametric Tests for the One-Sample Location Problem. Journal of the American Statisdcal Association . 77, 163-169. Dietz, E. J. (1984). Linear Signed Rank Tests for Multivariate Location. Communications in Statistics . 13, 1435-1451. Friedman, M. (1937). The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance. Journal of the American Statistical Association . 32, 675-701. 107

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108 Genz, A. C., and Malik, A. A. (1980). Remarks on Algorithm 006 : An Adaptive Algorithm for Numerical Integration over an N-dimensional Rectangular Region. Journal of Computational and Applied Mathematics . 6, 295-302. Greenhouse, S. W., and Geisser, S. (1959). On Methods in the Analysis of Profile Data. Psvchometrika . 24, 95-112. Hdjek, J., and Siddk, Z. (1967). Theory of Rank Test . Academic Press, New York. Hannan, E. J. (1956). The Asymptotic Powers of Certain Tests Based on Multiple Correlations. Journal of the Roval Statistical Society . Ser. B, 18, 227-233. Hodges, J. L. (1955). A Bivariate Sign Test. Annals of Mathematical Statistics. 26, 523527. Hollander, M., and Wolfe, D. A. (1973). Nonparametric Statistical Methods . Wiley, New York. Hora, S. C., and Iman, R. L. (1988). Asymptotic Relative Efficiencies of the RankTransformation Procedure in Randomized Complete Block Designs. Journal of the American Statistical Association . 83, 462-470. Hotelling, H. (1931). The Generalization of StudentÂ’s Ratio. Annals of Mathematical Statistics . 2, 360-378. Hsu, P. L. (1938). Notes on HotellingÂ’s Generalized T. Annals of Mathematical Statistics . 9, 231-243. Huynh, H., and Feldt, L. S. (1970). Conditions Under Which Mean Square Ratios in Repeated Measurements Designs Have Exact F-Distributions. Journal of the American Statistical Association. 65, 1582-1589. Iman, R. L., and Davenport, J. M. (1980). Approximations of the Critical Region of the Friedman Statistic. Communicarions in Statistics. 6. 571-595. Iman, R. L., Hora, S. C., and Conover W. J. (1984). Comparison of Asymptotically Distribution-Free Procedures for the Analysis of Complete Blocks. Journal of the American Statistical Association . 79, 674-685. Jensen, D. R. (1977). On Approximating the Distributions of FriedmanÂ’s % r and Related Statistics. Metrika . 24, 75-85. Joffe, A., and Klotz, J. (1962). Null Distribution and Bahadur Efficiency of the Hodges Bivariate Sign Test. Annals of Mathematical Statistics . 33, 803-807. Johnson, M. E. (1987). Multivariate Statistical Simulation . Wiley, New York. Kepner, J. L., and Robinson, D. H. (1988). Nonparametric Methods for Detecting Treatment Effects in Repeated-Measures Designs. Journal of the American Statistical Association. 83, 456-461.

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110 Wilks, S. S. (1946). Sample Criteria for Testing Equality of Means, Equality of Variances, and Equality of Covariances in a Normal Multivariate Distribution. Annals of Mathematical Statistics . 17, 257-281. Winer, B. J. (1971). Statistical Principles in Experimental Desien . McGraw-Hill, New York.

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BIOGRAPHRICAL SKETCH Show-Li Jan was bom in 1961, in Keelung, Taiwan, the Republic of China. In 1983, she graduated from the National Chiao Tung University in Taiwan, R.O.C., with a bachelor of science in applied mathematics. She then came to the U.S.A. and enrolled at Michigan State University at East Lansing, and received a master of science in applied mathematics in 1985. In the fall of 1985, she transferred to the University of Florida in Gainesville to pursue a doctorate in statistics. Ill

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ronald H. Randles, Chairman Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Cv-o Pejaver. V. Rac Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. "iWatX Eio-cA Louis S. Block Professor of Mathmatics This dissertation was submitted to the Graduate Faculty of the Department of Statistics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May, 1991 Dean, Graduate School