UFDC Home |   Help  |   RSS

 Group Title: distribution of Hotelling's generalized To² Title: The Distribution of Hotelling's generalized To²
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
 Material Information Title: The Distribution of Hotelling's generalized To² Alternate Title: Hotelling's generalized To² Physical Description: viii, 120 leaves. : ill. ; 28 cm. Language: English Creator: Hughes, David Timothy, 1943- Publication Date: 1970 Copyright Date: 1970
 Subjects Subject: Distribution (Probability theory)   ( lcsh )Statistics thesis Ph. DDissertations, Academic -- Statistics -- UF Genre: bibliography   ( marcgt )non-fiction   ( marcgt )
 Notes Thesis: Thesis--University of Florida, 1970. Bibliography: Bibliography: leaves 117-119. Additional Physical Form: Also available on World Wide Web General Note: Manuscript copy. General Note: Vita.
 Record Information Bibliographic ID: UF00097720 Volume ID: VID00001 Source Institution: University of Florida Holding Location: University of Florida Rights Management: All rights reserved by the source institution and holding location. Resource Identifier: alephbibnum - 000559442oclc - 13493516notis - ACY4898

PDF ( 3 MBs ) ( PDF )

Full Text

THE DISTRIBUTION OF HOTELLING'S

GENERALIZED T2

By
DAVID TIMOTHY HUGHES

A DISSERTATIION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1970

Dedicated to my offe Margaret and tom ae'

for their Zove and understanding

ACKNl~OWLEDG EVENTS

I would like to express my deep gratitude to

Dr. J. G. Saw~ for his constant assistance and encouragement.

Without his guidance, this work would not have been possible.

I would also like to thank Drs. P. V. Rao and

Richard L. Scheaffer for their careful proofreading and

helpful criticism. Special thanks must also go to Miss

Gretchen Uzzell for her excellent editing and to Mrs. Carolyn

Lyons w~ho turned a crude manuscript into a masterpiece of

typing.

iii

Page

\CK'NOWrLEDGEI EN TTS .................... .----------- iii

LIST OF TABLES .......................--------***-- vi

ABSTRACT . .....................-------------- vii

Chapter

I. STATEMENT OF THE PROBLEM .................... 1

1.1 Introduction ...................- 1
1.2 Historical Background .......1........ 6
1.3 Su~mm~ry of ResulYts ................... 10
1.4 HoItation, ........................- 11

II. ORTHOSPHERICAL INTEGR~ALS ................... 17

2.1 Introduction ........................ 17
2.2 Some Relationships on the
Orthospherical Integra~s ........... 20
2.3 Evaluation of OrthospherseaZ
IntegraZs .. . . . . . . 28

III. MIOMENTS OF: HOTELLING'S GENERA4LIZED To ...... 44

3.1 In!troduction *****-----*****........ 44
3.2 A Method for Evaluating the

3. 3 Th2e Fir~st Two Nom~ents of T ........... 56
3. 4 Evl~zuation~ of Certain Exupecta-tions

3.b5 Su~mmary .. . . . .. . - *

IV. PPOXIMATIONS TO THE DISTRIBUTION OF
H-OTELLINlG'S GENERA~LIZED To2 .....~,......... 75

4.2 IntrG~~odu to . . .. . . . 75
.2Avoroximations to th istribto
and Co~nvrison of Percentage
Points of T ........................ 76
4.3 Summary ...,....... ....... ........... 89

Chapter Page

V. THE POWER OF HOTELLING'S GENERALIZED T 2 .... 92

5.1 Introduction ......................... 92
5.2 An Upper Bound on the Integral

5.3 A Lower Bound on the Integral
of te Poer uncton o To ...... 99
5.4 The Monotonicity of the Power
Function of To2 .................... 110
5.5 Summary .. . . . . . . . 114

BIBLIOGRAPHY ........................................ 117

BIOGRIAPHICAL SKETCH ................................. 120

LIST OF TABLES

Table Page

2.1 Orthospherical Integrals .................... 39

4.1 Upper 5% points of T ........................ 83

4.2 Upper 1% points of T ........................ 84

4.3 Upper 5% points of T ........................ 85

4.4 Upper 1% points of T ........................ 86

4.5 Upper 5% points of T ........................ 87

4.6 Upper 1% points of T ........................ 88

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE DISTRIBUTION OF HOTELLING'S
GENERALIZED T 2

By

David Timothy Hughes

August, 1970

Chairman: Dr. J. G. Saw
Major Department: Statistics

Numerous criteria are available for testing the

multivariate general linear hypothesis. In this work some

properties of one of these criteria, T (a constant times

Hotelling's generalized T 2), are investigated. The T
statistic is the sum of the latent roots of a matrix obtained

from random sampling from a multivariate normal population,

and its distribution depends upon the corresponding popu-

lation roots and some degrees of freedom parameters.

The orthospherical integrals involve expectations

of the elements of a matrix having invariant measure on the

orthogonal group. These integrals are defined and a method

for evaluating them is presented. We have derived a special

representation of T which, with the aid of the orthospherical

integrals, enables us to present a method of evaluating the

moments of T. T'he first two nonnull moments are obtained

as an example.

vii

The exact distribution of T in the general case is

difficult to deal with and for this reason, three rather

simple approximations to its distribution are obtained.

All of these approximations are obtained by equating the

moments of T wiith the moments of the approximating random

variable. Accuracy comparisons of percentage points in

certain special cases indicate the usefulness of two of

these approximations.

A result by Wald and Hsu on the power function of

the analysis of variance test is used to obtain an upper

bound on the integral of the power function of T. A

lower bound on the integral of the power function of T

is obtained by using the properties of completely monotone

functions. Completely monotone functions are also used to

demonstrate the monotonicityv of the power function of T.

viii

CHAPTER I

STAiTEMENT OF THE PROBLEMS

1.1 Introduction

Suppose we observe a number of m-variatee column

vectors, X.:1=12..,k j = 1,2,..,n. obtained by
-13 1-
random sampling from each of k; m-variate normal populations

with common dispersion matrixi, V. Assume that the elemenlts

of the column vectors associated with population i: i=1

2,...,k, have expectations given by the elements of an

m-diensonalmea vecor,9..It is desired to test the
m-diensinal ean ecto, r1

hypothesis

H :ui uz **=u
O -r\

against (I.1.1)

H :at least one of the above qualities
a does not hold.

W~ith

N = r n. ( .1 2
i=1

define x. and x by

n.
1

- 1

3

and

k ni
x .
i=1 j=1

Construct the i?.x m matrix,

k ni
S2 = C (x..
i=1 j=1 1

- x. ~x..

-Y x. ]

(1.1.5)

where (*)' denotes the transpTose of the vector in brackets.

The matrix 52 will be nonsingular with probability one,

whenever S k >m (see [1] or [32]). An appropriate test

statistic for testing (1.1.1) is

k
To = (N k) n. x.
i=1

s Sp1(
- x ] S2~ ( .

-x

=(N k) tr S1S2-1,

where S1 is the m x m matrix defined by

k
S, = n. i.
i=1

- x ]x.

-x ] .

The a~ll hy-pothesis is rejected if

T > t,

where t is chosen to satisfy

(1.1.4)

(1.1.7)

(1.1.8)

P(To2 > tI0) = 2

(1.1.9)

3

for some desired level of significance, a. Thle quantity,

T 2, defined in (1.1.6), has the distribution of Hotelling's

generalized T 2

Mo~re generally, let ~i:i = 1,2,***,vi, be m-dimen-

sional, normally distributed column v\ectors w~ith m-dimen-

sional mean vectors, lq, and common m x m dispersion

matrix, V. Suppose S2 iS an m x m central Wishart matrix

(see [36]) distributed independently of Ez,**,[ with

dispersion matrix V and degrees of freedom v2 > i*

Then for testing the hypothesis

H : i ** = p 0,
O- --9

against (1.1.10)

Hl :at least one p. 0 ,
a -1

which is the canonical form of the general linear hypothesis,

an appropriate test statistic is Hotelling's generalized

T 2, defined by

To = 2 L q lS2 q~
i=1

=v, tr S1S2-1, (1.1.11)

where S1 is the mx; m matrix

V1

i=1

It is noted that S1 defined. in 1.12hain general, a

noncentral Wishart distribution with dispersion matrix, V,

degrees of freedom v, and m x m noncentrality matrix

i=1

(1.1.13)

In the first example given in which T w ras defined

by (1.1.6),

(1.1.14)

(1.1.15)

vl = k- 1,

V2 = N k,

and

1=~

(1.1.16)

wih e re

(1.1.17)

The general problem just given is invariant under

linear transformations. That is, if instead of observing

gi,***,Lv and S2!, w~e observe zz,***,z~ and S defined by
1 1

(1.1.18)

z. = v. : i = 1,***, v

and

S = AS2A'

(1.1.19)

- 5 -

for any nonsingulair :n i m natrix, A, the problem remains

the same. The m-variate~ column vet:ors. =;: i = 1, *, v,,

have mean vectors

(i = Ag. : 1 = 1,,v*, ( 1.0

and the hypothesis (1.1.10) is equivalent to

H :51 = 0,

against (1.1.21)

H :at least one 5 0.
a -i

This property leads to consideration of test

criteria which are also invariant under linear transfor-

mations. Any such~ criterion must be a function of the roots

of the determinantal equation

(S1 aS2| = 0, ( .2

where S1 is defined in (1.1.12). The distribution of such

a criterion is' a function only of m, v v, and the non-

zero solutions of

504I' XV| = 0. (..3

(At most, min(mn,v,) of AX,***,Xm are noncero.)

A number of test criteria have been proposed for

testing (1.1.10). These include:

-6

1. Likelihood ratio: H1+ i ;(1.1.24)
i=1

2. Roy's largest root: max(6 j; (1.1.25)

3.Pillai's criteria: 6 -; (..6
i=1

4C. Hotelling's generalized T :v 0. (.27
i=1

In this paper we investigate some of thie properties of

T = T 2/v2 ..8

for testing the general linear hypothesis. (It has become

standard practice in recent years to study the properties

of T rather thnan T_ itself. We will continue in this

practice and steadfastly- hold that the symbol "T" wJill

denote (1.1.28). The symbol "T 2" will be used only when

specifically discussing the properties of (1.1.11) rathier

than (1.1.28). The s?;mibol "tT' should not be confused Uwith

Hotelling's T2 Isee equation (1.2.1)). W~hen discussing

(T 2/v,) we shall use "(T)2" and reserve "T2" for (1.2.1).

1.2 Historical Backgo-und

In 1931, Hotelling [11] suggested a statistic which

he called T2 as a generalization of "Student's t" ratio.

Hotelling's Tz is in fact a special case of T given in

(1.111)when9, 1.Using (1.1.11), T2 may be defined as

- 7 -

T2 = vzy'S2-1~i (.2.1)

The distribution of T2 was derived by Hotelling [1l] under

the null hypothesis, and, in 1938, Hsu [14] derived the

nonnull distribution.

The statistic T (see (1.1.28)] was first suggested

in 1938 by Lawley [26] as a generalization of Fisher's z

(the familiar F ratio) for testing significance in multi-

variate analysis of variance. At that time, Lawley obtained

the distribution of

z=10{ my -2 1 To ') (1.2.2)

and showed that the limiting distribution of T as v2 tends

to infinity is the Chi-Square distribution on mvi degrees

of freedom. Hsu [15] also demonstrated this result and

obtained the null distribution of T in integral form. In

this same paper, Hsu derived the first two nonnull moments

of T, but, as w~e shall showz, his second moment is in error.

Somewhat later, Hotelling [~12] suggested T2 as a

generalization of his T2 statistic for measuring multi-

variate dispersion. In 1951, the null distribution of T *

was derived by Hotelling [13] for the case m = 2 and for

general v, and v; > 2 in terms of the Gaussian hypergeo-

metric function.

-

Since that time, numerous persons have studied the

distribution of T_ and T. Pillai [29] obtained thie first

three null moments of T aind used these to obtain an approxi-

mation to its distribution. Using this approximation, Pillai

and Samson [31] gave upper 54 and 1% points of T for m = 2,3

and selected values of vl and v2. In 1956, Ito [17] ob-

tained an asymptotic formula for the null distribution of

T_ as v2 + m and in 1960 extended this result to cover the

nonnull distribution [18]. Bagai [2] derived the null

distribution of T in terms of definite integrals for the

cases m = 2,3. The first tw~o nonnull moments of T 2

given by MIikhail [27] but his second moment, although it

differs from that found by Hsu, is also incorrect.

The distributions of latent roots of random matrices

were expressed by James [20], [22] and Constantine [3] in

terms of zonal polynomials [3], [21]. Constantine [4] used

these results to obtain the distribution of T, in both the

null and nonnull case, in series form. These series con-

verge, however, only for IT < 1. In this same paper,

Constantine derived the rth moment of T in terms of the

zonal polynomials for v2 > 2r + m 1 (the moments do not

exist otherwise) and listed the first two null moments

explicitly.

In 1965, Khatri and Pillai [23] derived the first

tw~o moments of T in the linear case (the case of only one

nonzero solution of 1.23 and the same authors used

- 9 -

these results to approximate the distribution of T in the

linear case [24]. They later generalized this to consider

any number of nonzero roots [25]. Most recently, Davis [6]

showed that, in the null case, the density of T satisfies

an ordinary linear differential equation of degree m of

Fuchsian type and obtains an equivalent first order system.

These results are used to tabulate the upper 5% and 1%

points of To2 1, fOr m = 3,4 [7]. Stein [34] showed that
H-otelling's T2 iS an admissible test of

Ho:p = 0,

against (1.2.3)

Ha:Fp 0 ,

and Ghosh [10] extended this by showing that T 2 iS admis-

sible for testing the general linear hypothesis. Das Gupta,

Anderson and Mvudholkar [5] showed that the power of T 2 iS

an increasing function of each X., the remaining X.'s being

held fixed, where Az,**,''m are defined in (1.1.23).

Mikhail [27] and Srivastava [33] later claimed that the

power of To2 depends upon X,,''!,[ only through

i=1

This claim is false, however, since the second moment of

T 2 depends upon

i=1

- 10 -

Ito [19] gave pow~ier comparisons of To2 and the likelihood

ratio criteria for the linear case and noted little or no

difference in their powers for large V2. For the case

m = 2, extensive power comparisons of T and three other

test criteria for the general linear hypothesis have been

made by Pillai and Jayachlandran [30].

1.3 Summary of Results

Before proceeding with a discussion of the

distribution of T, some preliminary mathematical results

are needed. Therefore, Chapter II is devoted to the

"orthospherical integrals." These integrals are defined

and a method of evaluating them is presented. The ortho-

spherical integrals up to order eight are tabulated at the

end of Chapter II. In Chapter III, a general method for

evaluating the moments of T is presented and the first two

nonnull moments are found explicitly. These moments are

used in Chapter IV to obtain three approximations to the

distribution of T. The 5% and 1% points of T are found

for each approximation, and accuracy comparisons are made

with the exact percentage points and Pillai's approximate

percent-age points for m = 2. In the case m = 3,4, compari-

sons ar~e made w~ith the percentage points given by Davis [7].

Upper and low~er bounds are given for the integral of the

power function of T. In addition we offer an alternative

proof of' the result due to Das Gupta, Anderson and Miudholkar.

- 11 -

1.4 Notation

Any work in multivariate analysis is greatly

facilitated by frequent use of matrix notation, and it is

also very convenient to abbreviate the distributional

properties of random variables. For these reasons, certain

notational conventions have been adopted. Although many of

these are standard, they will be listed here for reference.

1. Matrices will be denoted by capital letters and

the first time a matrix appears, its rowz and

column dimensions will be given. Thus A:(r x s)

denotes a matrix with r rows and s columns. The

matrix having all its elements zero will be denoted

by (0). "I" will be reserved for the identity

matrix.

2. The elements of a matrix will be denoted by the

corresponding small letter with subscripts to

denote their rowz and column position. Thus a..

denotes the element in the ith row~ and jth column

of the matrix A. The symbol [A)i is equivalent

to a.. and is somretimes used for convenience.

3. An underscored small letter invariably represents

a column vector. Its rowJ dimension will be given

the first time a vector appears. Thus x: [m) denotes

a column vector consisting of m elements. The

-12

vector which has all its elements zero will be

denoted by 0.

4. It is sometimes convenient to form row or column

vectors from the rows or columns of a matrix.

The symbol (A).~ will denote the row vector

formed from the ith row of the matrix A. The

symbol (A) denotes the column vector formed
from the jthz column of the matrix A.

5. It is sometimes convenient to partition a square

matrix AZ:(r x r) as

All I Al2

where All is the p x p matrix formed from the

first p rows and columns A; A12 is the p x q

matrix formed from the first p row~s and the last

q columns of A, p + q = r. A21 and A22 are simi-

larly defined.

6. The Kronecker product of A: (r x r) and B: (s x s)

will be denoted by A 8 B and is a matrix C:(rs x rs)

w~ith

c~k-1)s+i,(-1)s+j = akeib .I
7. If x is a random variable having the normal density

with mean y, and variance a this will be denoted

by

-13

8. If x is a random variable on (0,=) with densit)-

f(x) = {7( ]2(J-8/23- V1 xylx{1 21

we shall abbreviate this by

This is to be read as "x has the central Chi

density on v degrees of freedom."

9. If x is a random variable on (0,=) with density

f(xu) = exp -, -- {jr( A v]2(u+ 2j)/2 -
j=0

x ~jy exp [-7),

we shall denote this by

This is to be read as "x has the noncentral Chi-

square density on v degrees of freedom with non-

centrality paramneter, ."If X = 0, the density;

becomes central.

10. If xt and x? are independently distributed with

xtL ^ X, 2 9)
and

x2 ^ 7~ 22(0),

-14

then

z 2 X1
V1 X2

has the noncentral F density on vl and v2 degrees

of freedom and noncentrality parameter, A. This

will be abbreviated

11. If xl and x2 are independently distributed with

X1i ^ X. 2(A

and

x2 ^ X~ 2(0),

then

X2
x1 + x2

has the noncentral Beta density on vl and v2

degrees of freedom with noncentrality parameter,

A. This will be denoted by

u h Se(va~v2,A).

12. If xc is an m-dimensional column vector whose

elements have a joint normal density with mean

vector, i: (m) and dispersion matr-ix V: (m x m),

this will be denoted by

xr N (lcLV).

-15

13. Lets x**,xn be independent m-variate column

vectors with

x. lu.,Vi. : i = 1,*** n.

Then wit~h X = (xxl,***,xn and :: = (s**E)

the joint density of the elements of X will be

abbreviated

X mxn(V I)

14. With X an m x n matrix such that

X Nmxn(V I)

the m x m matrix

S = XX'

has a noncentral TWishart distribution with

dispersion matrix V, degrees of freedom n, and

noncentrality matrix 'Ii`I'. This wiill be abbre-

viated

S ~ Wm(V,n,?bml').

15. If S is an m x m symmetric matrix whose m(m + 1)/2

mathematically independent elements have the density

f(S) = K (V,v) IS (9-m-(V2P ex tr V-15)

on S positive definite with

Km(V,V) = {2my/2 m"(m-1)~YI/4 V v/2 'L
j=1

- 16 -

this will be abbreviated by

S Wm(Cv,(0) .)

This is to be read as "the m(m + 1)/2 mathemati-

cally independent elements of S have the central

Wishiart density with dispersion matrix, V, and

degrees of freedom v."

16. The symbols "a ==> b" and "a <==> b" are to be

read "a implies b" and "a implies, and is implied

by, b," respectively.

17. Suppose it is desired to integrate the function

fCxx,***,xn) over the set

A = {(x ,***,x ] : a. < x.
Then

f(x) d(x)

will be used to abbreviate

bi b
n f ( x t, *,x ). d x * * d x y
ai a

18. For referencing within the text [*] will denote

bibliographical refeyprencs while (*) will denote

references to equations. Thus [8] refers to the

eighth entry in the bibliography, while (4.2.6)

refers to equation six of Section 2 in Chapter IV.

Other special notational conventions will be

introduced as they occur in the text.

CIRAPTER II

ORTHOSPHERICAL INTEGR~ALS

2.1 Introduction

Let R(m x m) denote the group of m x m orthogonal

matrices. Let 4 be an m x m matrix of random variables

with 4 belonging to the group R(m x m), and suppose that 4

has invariant measure on R(m x m). This measure can be

determined in the following manner. Consider any row~ of 4,

say the first row for convenience. Then since 4 is orthogo-

nal, the elements, $1 : j = 1,***,m, of (4)1. must satisfy j=1 This is equivalent to saying that (0)1. determines a point on an m-dimensional hypersphere of unit radius. If 4 is to have invariant measure on R(m x m), then the measure on (0)1. must be fL(Q) 3 s~l1 m (2.1.2) = otherwise , where S. (r) = 2aj/2rj-1./T(j/2 (.13 - 17 - -18 is the surface area of a hypersphere of radius r in j dimensions. Now fix the elements in the first row~ of 9, and consider any other row~, say the second rowj for convenience. Due to the orthoo~nality of 4, the elements, O2 : j = 1, ***,m, of (0)2. mUSt Satisfy I$1]#2j = 0,
j=1
and (2.1.4)

j=1

This implies that (0)2. determines a point lying on an

(m 1)-dimensional hypersphere of unit radius which is

orthogonal to the vector (0)1.. Since 4 is to have invari-

ant measure on R(m x m), the measure on (b)2. COnditional

on (4) must be

fl( ~~ ~ )2 T1]=S 1(1 if (2.1.4) is satisfied;
m-1
(2.1.5)

= otherwise.

Continuing in thiis manner, fix the elements of

10)1,( 2.,**,()i-,.,and consider the ith row,()

of 0. Since 0 is to be orthogonal, the elements, 4 ..

j = 1,***,m, of (0). must satisfy

-19

j j ij =

j=1

j=1 i1j1

0,

0,

(2.1.6)

0,

and

j=1

This implies that (0)i determines a point on an (m-i+1)-di-

mensional hypersphere of unit radius which is orthogonal to

the mutually orthogonal vectors (4)1.,(4)2.,***,(4).
1-1,.
Since @is to have invariant measure on G2(m x m), the measure

on (0). conditional on (0)1 ,(4)2)** ). i 1
.' i-1,.

m 1+

if (2.1.6) is satisfied

; (2.1.7)

= otherwise.

Hence the measure on Q is

S1
f(0)= II g- (-
]=1 j

if QQ' = I;

(2.1.8]

= otherwise.

f (4). 10)z.,(s)2.y
1.|

-20

Using (2.1.2), we can write the measure on 4 as

j=1 2,j1

= otherwise.

(2.1.9)

In the derivation of the moments of T, we shall

need certain moments and product moments of the elements

of 0. To this end, for any m x m matrix, C, of nonnegative

integers, define

m m c..
J(C) =!n n: H H
ij
i=1 j=1

(2.1.10)

These are the "orthospherical integrals." We shall say

that J(C) is of order s, where

m m
s = C c...
i=1 j=1 II

(2.1.11)

2.2 Some Relationships on the Orthospherical Integrals

Let P and Q be fixed m x m orthogonal matrices and

define an m x m random matrix, 0, by

o = P~QQ,

(2.2.1)

m m c..
= H H 0 1 f(0) de.
I n(mxm) i=1 j=1 i

- 21 -

wherein 4 is random wdith invariant measure on R(m x m). Then

GeR(m x m) and, since it is obtained from Q by an orthogonal

transformation, O also has invariant measure on. R(m x m).

Now~

m m c..
J(C) =Ir ET H .. 11
i=1 j= 13

m m m m Ic
= Ei H H iaaB (2.2.2)
i=1 j=1 a=1 B=1

By suitable choice of P and Q, (2.2.2) yields a number of

interesting relationships on the orthospherical integrals.

In particular, suppose that P is obtained from the

m x m identity matrix, I, by some permutation of the rows

or columns of I, and that Q is similarly obtained by a

possibly different permutation. P and Q are therefore

orthogonal, and clearly they have one and only one element

in each row and column equal to unity and all other elements

zero. That is,

p.a = 1 if al = i. for some i.

= if a / i., (2.2.3)

where i. differs for each i, and

q.=1 if B = j for some
83 j j

= if 6 f j; (2.2.4)

-22

*
where jj diiffers for each

j. Now~

m m
Z Pi/aB 9j
a=lf1 1

(2.2.5)

1j

so that (2.2.2) can be written as

c..j

(2.2.6)

#- a
1. ,j
1 j

with 4 the elements of 6, where 6 is defined in
i. ,
Sj
(2.2.1). e may rewrite (2.2.6) as

(2.2.7)

with c. ", the elements of C*:(m x m) defined by
i j

(2.2.8)

C* = P'CQ'.

That is, C* is obtained from C by a permutation of the rowl~s

or columns of C. Since by definition

J(C*]=E 4,
i=1 j=1m m i Ci ]

(2.2.9)

then

(2.2.10)

J(C) = J(C*;).

The permutation matrices, P and Q, were obtained arbitrarily

Thus, for

so that (2.2.10) holds for all permutations of C.

example, with m = 4

m m
J(C) =t E H
i=1 j=1

J(C) =E H H 4 1,
i = 1 j=1mr

- 23 -

2 01 0 0 00 0
0 0 00 0 3 00
0 3 00 1 00 2
0 0 00 0 00 0
3 00

I O1 OO OO2 0OO

and so on.

Equation (2.2.8) can be used to simplify the
notation by displaying only the smallest rectangular array
containing all of the nonzero elements of C, along with an
index, m, to denote the dimensions of 4 and C. If C is the
smallest rectangular array containing all nonzero elements
of C, then

J(Cc) = J(C). (2.2.11)

For example, with m = 4,
2 00 1
S21 0 0 00 0
0 030 03 0
0 0 0 0

- 24 -

and so on.

Let P and Q be diagonal matrices whose diagonal

elements are either +1, so that P and Q are orthogonal.

Then from (2.2.2) and (2.2.11),

J (C) = p 3 1
m _r (ii jj) ij

i= =m m c I

=J (C) HI HI (p q ]
mi=1 j=1

m C. m C.
= J C Hp.'Hq (2.2.12)
i=1 3=1.

1i. 1]
j]-1I

and (2.2.13)

c c...
*)i=1 13

Suppose that ci. is an odd integer for some iEC1,***,m).
For convenience, assume that cl. is odd. Setting

pii = -1,

pi = +1: i = 2,***,m, (2.2.14)

q. 1: j = 1,***, m,

-25

equation (2.2.12) gives

Jm(C) = -Jm(C). (2.2.15)

It follows that

Jm(C) = 0 (2.2.16)

whenever any row~ or column of C sums to an odd integer.

Thus, for example,

J =I 0,

and

J2 4 2 0.

By considering other special matrices, P and Q,

many other relationships can be obtained on the orthospheri-

cal integrals. For example, choosing

1 1

1 1

(0) I

and (2.2.17)

Q =I,

- 26 -

equation (2.2.1) yields

and (2.2.18)

622 = -(12 ~22 *

Hence

2 0 = E(61128222)

= ES(011 + ~21 2 912 "2

(2.2.19)

Expanding (2.2.19) and making use of (2.2.16) gives

2 J = J (,2) J (2.2.20)
15m 11 LO 2m27

Thus J,1 ( ) can be evaluated once J (2,2) and J (2 0)ar
known .

Other relationships can be obtained by simply

making use of the orthogonality of Q. Some examples are

presented below.

i.~12 = 2 2 j
j=1

=~~ i + 1 $11$1. (2.2.21)
j=2

-27

Taking expectations in (2.2.21) yields

Jm(2) = Jm(4) + (m 1) Jm(2,2).

i=1

= Oil 12~ +11~12 ~i2 .
i=-2

(2.2.22)

(2.2.23)

Taking expectations yields

J (2,2) = J (2,4) + (m 1) J (2 2).

(2.2.24)

3. O = C q2
j=1

~11 21 1 2'.l
j=1

m
= 11212 21 ~ 11 21 12.
j=2

(2.2.25)

Thus

Jm(2,2) + (m 1) Jm( : ) = 0.

(2.2.26)

Such relationships greatly simplify the evaluation

of the integrals, since they reduce the number of integrals

which must be evaluated directly. Direct evaluation of the

orthospherical integrals can be tedious, but a relatively

easy method is presented in the nexct section.

- 8

2. 3 Evaluatic~z on 3- rthoschearical rinteg7ra~

The orthospherical integrals which need to be

evaluated directly may be evaluated by a number of methods.

A polar transformation on the elements of Q may be used

when the integrals involve only the elements in a single

row~ or column of 9. This method becones extremely cumber-

some, however, wLhen elements of more than one rowd or column

are involved. Perhaps the simplest method is one which

depends upon the following representation of 0.

Let X be an m x m matrix of random variables and

suppose

X Ynxm((0),I 0 I).(2.1

Define an m x m matrix R to be a lower triangular matrix

satisfying

XX' = RR'. (2.3.2)

Equation (2.3.2) can be writteni as a system of equations
onth elemennt of' I and R.

2 2
x2 = ral
j=1

x ,Ixi = r"iril: i = 2,***,m,
j=1

2_2 r22
x =r21 i2
j=1

-29

j 1 xi = r21ril + r22ri2: i = 3,***,m,

and
m m

j=1 m3 j=1 m)

If the diagonal elements, r i: i = 1,***,m, are required

to be nonnegative, then the equations (2.3.3) uniquely

determine the elements of R. Hence, R is the unique lower

triangular m x m matrix with nonnegative diagonal elements

satisfying (2.3.2). We shall call R the lower triangular

square root of XX'.

Define a matrix 9:[m x m) by

X = RO. (2.3.4)

It is to be noted that the elements of X, and hence of R,

are continuous random variables. Therefore, the diagonal

elements, 'i : i = 1,***,m, will be positive with proba-

bility one, and hence R will be nonsingular with probability

one. Then 4 is properly defined by (2.3.4) and

4 = R- X. (2.3.5)

Using (2.3.2) and (2.3.5), O must satisfy

94' = (R- X)(X'R' ')

- 30 -

=R 1(XX')R'-

=R- (RR']R'-

=I, (2.3.6)

so that a has zero measure except on a subset of G(m x m).

Let f(X) denote the joint density of thie elements of X

described by1 (2.3.1), and P be any fixed, orthogonal,

(m x m) matrix. Since f(X) > 0 for any real X, then

f(XP) > 0. From (2.3.2), and the fact that P is orthogonal,

w~e have

RR' = XX'

= XPP'X'

=(XP)(XP)'. (2.3.7)

Thus the lo-wer triangular square root, Ri, of XX' is identical

to that of XPP'X'. Then

XC = RO ===> XP = R3P. (2.3.8)

This implies that whatever has nonzero measure, then so

does OP. If g(0) is the measure on then

g(0) > 0 for all OEd(m x m). (2.3.9)

The matrix O will be shown to have invariant

measure on ;l(m x m). The joint density of O and R will be

obtained, and it w~ill be shown that the elements of Ri are

oothi iutua~lly innde deni~t; and independents of thle elements

31 -

of d. The joint~ density of R and O Iwill be denoted by-

h(R,Q). This joint density can be obtained using the

density of X given by (2.3.1), and obtaining the Jacobian

of the transformation from X to R and O defined by (2.3.4)

and (2.3.6). Denot~ing this Jacobian by- J(X R,Q), then

h(R,0) = f(RO) J(X +t R,O), (2.3.10)

where f() is the density of (32..1). Direct evaluation

of the Jacobian is cumbersome and the method used here is

due to Hsu and reported by Deemer and 01kin [8].

We seek the Jacobian of the transformation from X

to R and Q defined by

X = RO,

and (..1

QQ' = I,

with all matrices m x m and R lower triangular. Let gX,

6R, and 60 be n x m matrices of elements, (6X) j, (6R) ,'

and (60) j, denoting small changes in the elemensofX
R, and Q respectively. Suppose that the changes bR in R

and 60 in 4 bring about a change dX in XY so that (2.3.11)

is preserved. That is,

X + dX = (R + 6R)(Q + 60)

and (2.3.12)

(4 + 66)(8 + 60) = I.

- 32 -

Expanding equations (2.3.12) and dropping terms of second

order in the 6's, we find thiat

X + 6X = RG + R(60) + (8R)

(2.3.13)

anid

Making use of (2.3.11), w\e can write equations (2.3.13)

as

aX = R(60) + (6R) 4

(2.3.14)

and

@(60)' + (60) O' = 0.

Hsu has shown that

J(X +t R,0) = J(6X +t 6R,64),

(2.3.15)

wJhere J(6X; + 6R,64) is the Jacobian of the transformation

from 6X to 6R and 60 defined by equations (2.3.14), in

which R and 4 are now.~ considered to be fixed m x m matrices.

For convenience, denote 6X by A, 6R by B, and 64

by C.

Then

J(X +t R,4) = J(A +t B,C),

(2.3.16)

with

A = RC + Be

and

(2.3.17)

CQ' + QC' = 0.

Define m x m matrices A* and C* by

A*: = AQ''

and (2.3.18)

C* = CQ'.

Using (2.3.17) and (2.3.18),

A* = RC*k + B

and (2.3.19)

C* + C*' = 0,

so that C* is an asymetric matrix. Furthermore,

J(A +t B,C) = J(A + A*) J(A* +t B,C*) J(C* +t C).

(2.3.20)

The Jacobians, J(A +t A*) and J(C* C), are the Jacobians

of the transformations defined by (2.3.18). These are

orthogonal transformations and hence,

J(A +t A*) = J(C* +t C) = 1.

(2.3.21)

The Jacobian J(A* *t B,C*) isthe Jacobian of the transfor-

mation defined by (2.3,.19). Recall that B = R, and hence

is lower triangular. Direct evaluation of this Jacobian

can be carried out rather easily. It is in fact

J(A* *t B,C*)

8( x~ 21a 2 ** a ** a *ax*a 3 ** *a2
mlm2 mm Im

... a C...2Ja22 *m ., a2a3.. 2
...9 h...am lmf (2.3.22]
r2m 'm-1, m

GC

> 10 i .~ . . .
CO *I *i *

I *
OC *M O I

e I

3 *C I **
CO I h h
-r l oi o o .. .0 .
N sH

~ E I
O I

*M

* I r
*M~I O
ON

O i

*H
*-

CD E

O -

I R * K*

,0 * * * i U 0 * U U *

-34

where L is an m(:2-1)/2 x m(m + 1)/2 matrixe which is

irrelevant to the value of the determinant of the above

matrix, since the corresponding upper right hand partition

of the above aratrix has all its elements zero.

The Jacobian J(A* B, C") is therefore

m-
J(A* B,C*) = Hr..(23.3
i=1

Equations (23.516), (.3.2.0), (2.3.21) and (2.3.23) imply

m-i
J(X +t R,0) = Hi r. 2.3.24)
i=1

The joint density; of R and Q is then obtained from (2.3.1),

(2.3.10) and (2.3.24).

-m2/2 1 t R m m-i
h(R,Q) = (2x) exp y tr RR Hq

on

~EO!a x m), (2.3.24)

r.. > 0: i = 1 *m

and

-m < ri < +m: i = 2 ,**,m; j = 1,***,i-1.

This density factors into

m m i-1
h(R,Q) = h1(3) Hh.(rj= ] h.r
i=1 = =

(2.3.25)

- 36 -

hl(@) = j 1 on @ER(m x n),

(2.3.26)

m-ii-1 2 lm-i 1
hi(r..) = Ir( ]i 2 r.exp -- r..

on r.. > 0,

and

h..( I = 2)ep 1 r.." on -m < r.. < +m.
1113 i 3

Thus 0 has invariant measure on n(m x m), and its elements

are independent of the elements of R. Furthermore,

r.. ^ (0): i = 1,***,m,

and (2.3.27)

r.. ( ) i = 2,***,m; j = 1,***,i -1.

Using (2.3.4) along with the distributional

properties of X, R and 4, the orthospherical integrals

can be evaluated. Some examples follow.

1. xii = raz~11. (2.3.28)

Hence,

E(XII ] = E(rix 011 )

= Er112 ) J(2). (2.3.29)

(2.3.1) and (2.3.27)

Equations

imply that

E(X112)

and

(2.3.30)

E(r112 m

Substitution into (2.3.29) yields

1
J (2) -

(2.3.31)

2. E(x11 ) = E(rix 11 )

=E(r11 ) E(011 )

= E~r11 ) Jm(4).

(2.3.32)

But

E(X 1 ) = 3

E(r11 ) = m(m + 2),

and

(2.3.33)

implying that

3
Jm(4) =mm+2.

(2.3.34)

(2.3.35)

3. X12 = 11112.

From (2.3.28),

E(XII2X122) = E(r11 Oil #122),

(2.3.36)

and hence

3S

E(x11 ) E(X12 ) = E(rix ) E(011 412 )

= E(r11 ) Jm(2,2).

(2.3.37)

Since xll and x12 are independently distributed as N(0,1),

1
Jm(22 m(m + 2)

(2.3.38)

The value of Jm(2,2) could have been obtained from
the relationship given by (2.2.22). WIe have

Jm(2) = J (4) + (m 1) Jm(2,2).

(2.3.39)

Substitution of (2.3.31) and (2.3.34) into (2.3.39) yields

9 = m(m + 2) + (m 1) Jm(2,2),

J, (2,2) 1 --Z.

(2.3.40)

(2.3.41)

so that

Other orthospherical integrals may be evaluated

in a similar manner. The orthospherical integrals up to

order eight, necessary in order to obtain the first four

moments of T, are contained in Table 2.1.

(The integrals up to order eight not included in this
table are equal to zero)

-39

TABLE 2.1

Orthospherical Integrals

A1 = m (m +

a, = m(m +

A 3 = m (m +

2) (m + 4)

2) (m + 4) (m + 6)

Order 2:

1
J (2)

Order 4:

J (4) 3~i

1
J (2,2) -1

_(m+1)
(m 1) 6

(m- 1) A

J(2 01
J)( :

Order 6:

15

J (4 ,z~ 2)

-40

TABLE. 2. 1--Cont~inued

3 (m+ 3)
~(m- 1~) 2

3
(m-1) 6

1
62

Jm( a _

Jm( o 2

(at~~z

J(2 2' mi
mo 2 (m-1]62

(m2+3m-2)
S(m-2) (m-1)n2

S(m-1) 62

=- (m+ 2)
(m-2) (m-1) A2

2
S(m- 2) (m- 1) a2

Order 8:

105
As

Jm s

J (6, 2)

Jml )

-15
A 3

_ 5(m+5)n

I2 0 0 j
002

J I 2O 11 1

011

/110 j

011

-41

TABLE 2.1--Confir~ued

Jm(4,4)=

Jm4 o) 9(m+3)(m+5)
me 4 (m-1)(m+1)A3

J (4 22) ~mi
(42 3(m+ 1)

2,( _) 3(m+3)
mo 2 (m- 1) a3

S3(m+5)

= 3(m+3)(m+5)
S(m-1)(m+1)a3

3(m3+8m +13m-2)
= m-2)(m-1)(m+1)n3

(m-1)n,

-- 3(m +7m+14)
[m-2)(m-1)(m+1)n3

(m-1)A3

J, O 2O 2

J (o 4O2

I4 0 0 I

J/0 0 2

moI 31 3

- 9(m+3)
(m-1)(m+1)A3

Jm(o (m-1)A

I~

-42

TABLE 2.1---Continued

_3(m+ )

m, 31 1

= 3(m2+5m+2)
S(m-2) (m-1) (m+1)A3

J (2,2,2, 2) ~~

2 2 2 mln

m 0 0 3

J(2 2) (m +4m+15)
m2 2 (m-1)(m+1)n3

(m+3)2
S(m- 1)(m+1)n3

J 2O 2

J 2O 2

0 0 (m+j) (m+5)
2 2 (m-1) (m+1)a3

2 2 0 (3 62+m6
J
ma 2 0 -(m-2) (m-1) (m+1l)n3

22 0) (m+3) (m+5)
Jmioo 2 =(m-1) (m+1)n3
to a 2

2 2 0 0 (3+8m2+13m-2)

J -[00

o 2 co0 (m+3)(m3+4m -11m-2)
0 0 0 2 m3~-~7r~--'7i-~

3 Io
J 0
m 1 10

- 43 -

TABLE 2.1--Continued

1 m
( -m-1)(m+1)n3

J, 2 1

Jm 2O 2

_ 1
(m-1)63

(m2+7m+14)
S- (m-2)(m-1)(~m+1]63,

- (m2+3m+6)
S(m-2) (m-1) (m+1)n3

- (m+3)
S(m-1)(m+1)63

(2+5m+2)

(m+3)
S(m-1)(m+1)a3

(m3+6m2+3m-6)
(m- )(m- 2)(m- 1) m+1) 63

(mnJt-1) m1)6

Jmo 2O O

1 1
1 1

Jm( 11

(m +6m+16)

2 2 00

J = 01
mI0 I
1 201

2 0 11

J.[ 0

mooI 20
0 0 11

1J~ 1 111

1~ 1 0 O
80011

CHAPTER III

M\OMENTS Of HOTELLING'S GENERALIZED T_

3.1 Introduction

In Chapter I we defined Hotelling's generalized T 2

(see equation (1.1.11)) as

v1
T 2 = 92 r v.S,'v
O -- 1i-
i=1

= 2 tr S S2-1,

where, under normal assumptions on the underlying random

variables, SI and S2 are n x independent random Wishart

matrices with

and (3.1.2)

S2 ami, 2/0) .

It is assumed_ that T7 is positive definite. In view~ of the

invariance of the general linear hypothesis and T 1 under

linear tranisformations (see Section 1.1), w~e may simplify

the derivation of the moments of T "

Since i' is positive definite and

i=1

- 44

- 45 -

(see equation (2.1.13)) is nonnegative definite, there

exists an m x m matrix, G, satisfying

GVG' = I

and(.1)

GMM~'G' = A

where A is an m x m diagonal matrix with diagonal elements

X1,'**,Xm defined in (1.1.23). W~e may therefore assume,

without loss of generality, that in our definition of T *

given by (3.1.1), the column vectors q;: i = 1,***,vl and

the m x m matrix S2 are independently distributed with

7. N (5 I): i = 1,***,v l, (3.1.5)
and

S2 W m(I,v2,(0)j, (3.1.6)

where ii: i = 1,***,,v stisfy
-11

i.i = A.(.1 7
i=1

W',rite

Y = ( x,** ,v ](3.1.8)
and

so that Y is an m x vl ma*trix of random variables and Z is

an m x us matrix of means, and

-46

Y Nmx zE I0 I .( .0

From (3.1.1) we may write

T = V2 r v 'S2-lv
i=1

=V2 tr Y'S2- Y. (3.1.11)

In this chapter, a method is developed for evaluating

the moments of T 2, or,more precisely, the moments of

T 2

under the assumptions (3.1.6) and (3.1.10) with T_ defined

by (3.1.11). The first tw~o moments, E[T] and E[(T) ] are

found explicitly. It is noted that the rth moment of T

exists, provided v2 > 2r +t m 1.

3.2 A Method for Evaluating the Moments os T_

We have

T
V2

=tr Y'S2-1Y,(. .1

with Y and S2 distributed as in (3.1.10) and (3.1.6). Let

dl,***,dm be the solutions of the determinantal equation

1S2 GII = 0 (3.2.2)

so that di,***,dm are the latent roots of S2. Define D to

be an n x m diagonal matrix ,ith

- 47 -

D = dl 0 * 0 (3.2.3)

0 d2 * 0

0 0 * d

If R. is an m-dimensional random, normed eigenv;ector associ-

ated with d., then

LS2L' = D, (3.2.4)

where L is the m x m random matrix satisfying

L = (1,***,E ) (3.2.5)

and

LL' = I, (3.2.6)

so that L is orthogonal. L, and D3 are uniquely defined by

(3.2.4), (3.2.5) and (3.2.6) if we require dl > d2

> dm and the elements of 1j: j = 1,***,mof the first
row of L to be positive. The joint density of L and D

will be obtained, and it will be shown that the elements

of L are independent of the elements of D. The Jacobian,

J(S2 + L,D), of the transformation from S2 tO L and D will

be obtained by the method used in Section 2.3.

Let 6S2, bL, and dn be m x m matrices of elements

(6S2) ,, (SL)i and (gD)i.. representing small changes in

- 48

the elements of S2, L, and D respectively. We require that

(L + 6L] remain orthogonal and(D + 6D) remain diagonal.

Suppose that the changes 6L, in L, and dD, in D, cause a

change 6S2 in S2 SO that (3.2.4) and (3.2.6) are preserved.

Then

S2 + 6S2 = (L + 6L)'(D + 6D)(L + 6L)

I = (L + 6L)(L + 8L)' (3.2.7)

By expanding (3.2.7) and dropping terms of order greater

than one in the 6's and using (3.2.4) and (3.2.6), wie have

6S2 = L'D(6L) + L'(6D)L + (6L] DL,

and (3.2.8)

0 = L(6L)' + (6L)L'.

As in Section 2.3

J(S, -t L,D) = J(6S2 + 8L, 6D) (3.2.9)

where J(6S2 + 6L, STD) is the Jacobian o'f the transformation

from 6S2 to 6L and 6D defined by (3.2.8), in which L and D)

are now assumed to be fixed m~x m matrices. For convenience,

denote bS2 by A, GL by B, and 8D by C. Then from (3.2.8)

and (3.2.9] we seek the Jacobian J(A4 + B,C) defined by

A = L'DB + L'CL + B'DL

and (3.2.10)

0 = LB' + BL'.

- 49 -

Define m x m matrices A* and B* by

A* = LAL'

and (..1

B* = B,'.

Using (3.2.11), equations (3.2.10) may be written as

A* = DB* + C + B*'D

and (3.2.12)

0 = B*' + B*,

so that B* is an asymmetric matrix. Then

J(A +t B,C) = J(A +t A*) J(A* +t B*,C) J(B* *t B) (3.2.13)

where J(A4 + A*) and J(B* B) are the Jacobians of the

transformations defined by equations (3.2.11), and

J(A* +t B*,C) is the Jacobian of the transformation defined

by equations (3.2.12). Equations (3.2.11) represent

orthogonal transformations and hence

J(A +t A*) = 1

and (3.2.14)

J(B* *t B) = 1.

Equation (3.2.13) then reduces to

J(A +t B,C) = J(A* + B*',C).

(3.2.15)

- 50 -

Direct ev-aluation of J(A*' i B*,C) is straightforward.

Recall that C = SD and hence is diagonal, B" is asymmetric

and A* is svaretric.

a(all~a22****a *a21*ast~as****a *~a
J(A* *t B* C) =mm ml m2
3(c 1 c 22 --- c b 2 1*b a "b 32****b *b *
mm ml m2

...a *)
...b *) (3.2.16)
m,m-1

Laying out the matrix of derivatives in tubular form, we

have

I

-51

So o ***C o

I

CO O

Ocl

0 0

***D

UCA i:i

***D f

U U

- 52 -

The Jacobian J(A*" B*',C) is the absolute value of the

determinant of the above matrix and is

Hi (d. d.].
i

J(A* B*,C) = H

(3.2.17)

Thus, from equations (3.2.9), (3.2.15) and (3.2.16),

Hm (d. d.].

J(S2 + L,D) = H
1

(3.2.18)

From (3.1.6), the density on S2 is

j=1

x S2r-2-m-8/2

(3.2.19)

on S2 positive definite.

The joint density of L and D,

g(L,D), is, from equation (3.2.4)

g(L,D) = f(L'DL) J(S2 + L,D)

(3.2.20)

where f(*) is the density of (3.2.19). Recalling that L

is orthogonal, we find that

g(L,D) = 2 92/2 .m(m-1)/4 m pv
3=1

x H d (2-m-1/2
j=1 3

ex (- d-) H (d.-d .]
3
(3.2.21)

ep t S2

-53

on L satisfying LL' = I, R,j > 0: j = 1,***,m and dl > d2
> *** > dm > 0. The density of (3.2.21) factors into

g(L,D) = gl(L) g2(D)

(3.2.22)

where

j=1

if LL' = I, RA > 0: j = 1,***,m

= otherwise,

(3.2.23)

and

.'"/2

exp(- d. H d -.
2 1l

if dl > d2 > *** > dm > 0

= otherwise.

(3.2.24)

Hence the elements of L are independent of the elements

of D.

Suppose that K' is any m x m random orthogonal

matrix satisfying

KS2K(' = D.

(3.2.25)

Write

( ki,k 2, ***, k ]

(3.2.26)

821D) = m22 H r( 2 t
j=1

x H d.(2-m -10/2
j=1 3

-54

whee k is an m-dimensional, random, normed eigenvector

associated with d-. Comparing (3.2.26) with (3.2.25), we

have

1
P~lk. = i 1. 2 :

(3.2.27)

and

P(ki -i] = i=

The measure on K; is obtained from

1,***,m.

(3.2.23) and (3.2.27).

It is in fac-t

m.2
2 =

if KK' = I

(3.2.28)

= otherwise

so that K has invariant measure on n(m x m) and is indepen-

dent of D.

With Y defined by (3.1.10) and Kt defined by (3.2.28),

let Z be an n x vi matrix of random variables satisfying

(3.2.29)

Z = KY.

From (3.1.10) the distribution of Z, conditional on K;, is

ZIK ^ Nmy (KE,I O I).

(3.2.30)

The unconditional density on Z can be obtained from (3.2.30)

and is-

- 55 -

f(Z jKER(mm) -mvl/3 7c 1
f(Z) =(2x) ~ x 2 tr(Z-K >)(Z-KE)'

x dK. 3.2.31)
j=1 2xr

It should be noted that since Y and S2 are independent, Y,

Ki and D are mutually independent. This implies the inde-

pendence of Z and D.

Using equations (3.2.1), (3.2.25) and (3.2.29), we

have

T = tr Y'S2-1Y

=tr(Z'K)(K'D-'K)(K'Z)

=tr Z D-'Z

m vl z..
= (3.2.32)
i1= 3=1 1

Wrri ting

V1
u. 2 =Z..2,
S i=1 1

m u.2
T = C.
i=1 1i

(3.2.33)

then

(3.2.34)

This representation of T, along with the results of

Chapter II, and the distributional properties of Z and D,

can be used to evaluate the moments of T.

- 56

3. 5 The FSirst Two Mlomen~ts of T

Using the representation of T given in equation

(3.2.39) the first twio moments of T are

E (T) = C E d:
1= 1

and

m U.2UiU.

i=1 di fj"1

= ~ E~ t + -- liu~) F
i= d Ir did

(3.3.2)

From equations (3.2.30) and (3.2.33), u.2 COnditional on
K is the sum of vl independent normal random variables.
Thus

u.2 K I^ X (KA').. : i = 1 m (3.3.3)

where from (3.1.7)

57-

A = EE'.(3.3.4)
For convenience, let

Y. = (KAK')..

= k.. A.i (3.3.5)
j=1 1
since A is diagonal. Then

u 2K ^ X 12 Y ]: i = 1,2,***,m. (3.3.6)

Furthermore, conditional on K, u.2 and u.2 are independent.
Hence, using (3.3.6)

E(u =) E E (ui2
K u-2 K

= Elv1 + Y ), (3.3.7)

E(U = E E (ui4
K ui2 K

=EK va(vi + 2) + 2y.(vl + 2) + y'n (3.3.8)

and

E(u 2u. = E E Iu u.21
SK iu.,uIK 13

=E E (u.2 E iuj
K u. K u. K

=EI vz' +( vj Y. 7 Y. (3.3.9)

-58

Recalling that K; has invrariant measure on- R(m x m), w~e may

use the results of Chapter II to evaluate the moments of

yi and the product moment, EKy(viY ] where Yi is defined in
(3.3.5).

E (y i) =E k..2X
K' K j=1 11 3

=J (2) X.

= A, (3.3.10)
mj=1

E(y2 => E k. 2Xj
K K j=1 3

= Ei 4X k." +C~i~i k. k .2
Kj=1 133 1 3

=J (4) 1 .2+J (2.2) A~ R .
m j=1 3 mE

j=1 m1+~ f
and
m m

= ~ i E .2 k 2 2;
K . =1 1 l 1 t i Et

-59

1 (m+1)
mTmTI A 1a T m-1)~nm~m+2 RR

Substitution of equations (3.3.10), (3.3.11) and (3.3.12)

into (3.3.7), (3.3.8) and (3.3.9) yields

i=1

(3.3.13)

E~u] =vl(vl + 2) + (2vl + 4) 1 X
j=1

+3 2 1 LA xxj
m(m+2) j m(m+2) j
j=1 i j

(3.3.14)

and

m
( 2v X.'
1 3 j=1

+ ~m+1~inZ Z .. Z ij
(m-1m~m2) 1

(3.3.15)

Upon substitution of (3.3.13), (3.3.14) and (3.3.15) into

equations (3.3.1) and (3.3.2), the first two moments of T
are

]

(3.3.16)

and

m m
F(T) = i + 1 E C
=1 1=1

-60

i=1

+ m-
+~+2 mC .j2 + 1 I
m~+)j=1 3 im2)1

m m
x c" 1 2 ] I 2vi X
i=1 di j=1

1 m+1
m(m+2) 1 =i 1

x d1 (3.3.17)

Evaluation of the terms

m m
=1 1i1 1

and

involve some special problems. TThese w~ill be dealt w~ith

separately- in the next section. W~e list these expectations

here, however, in order to obtain the mnoments of T:

E =v m1 (3.3.18)
i=1 1

-61

m(v2-1)
(V2-m)(V2-m-1)(v22-3-)

(3 >.19)

and

S1 m(m-1)
c:7 [v ) 2- .
ir)1

(3.3.20)

Substituting these expectations into equations (3.3.16) and

(3.3.17), we obtain

myi + A.X
B(T) = 1(3.3.21)
v2-m-1

and

=mv12(my2 m2-2m+2) + 2my (v2-1)

+ (2vl(my2n2-2_~2) 2 ~1-~j
i=1

m

i=1

(~.3.23)

Numerous checks are available on the moments of T.

The distribution of T, and hence its moments,

in the following special cases. If m! = 1,

is known

T ^ '
x yo)
L2

(3.3.23)

m

-~ d 2
1 1

i=1

- 2

Putting m = 1 in equations (3.3.21) and (3.3.22), the first

two moments of T reduce to the first two moments of a ran-

dom variable distributed as in (3.3.23). In the case of

v1 = 1, T2 reduces to Hotelling's T' defined in (1.2.1).

Hence when vl = 1,

T = 7.

(3.3.24)

The distribution of Hotelling's T2 iS knOwn to be [14]

V2X 2IX
T2 h
X 2 *

(3.3.25)

It should be noted that when vi = 1, there is at most one

nonzero solution of

A1 XII = 0.

(3.3.26)

Then w~hen vl

x 2_ (0)
V2 e

(3.3.27)

Putting v? = 1 in equations (3.3.21) and (3.3.22), we see

that the first twio moments of T reduce to those of a random

variable distributed as in (3.3.27). The limiting distri-

bution of T 2, as V2 tends to infinity, is known to be

(see [15] or [26])

lim T 2
o
\12+m L

(j3..28)

f
1 E
X 2 1. .
my 1
1 1=1 o

63

In equation (3.3.21), if we multiply byI v2 and take the

limit we obtain

lim
i m)-O T =' mvl + mC ..i (3.3.29)

Similarly, multiplying (3.3.22) by v22 and taking the

limit, we have

E li T =m2 2+ 2mvi + 2(muz+2) m. + 1
v2+im o =1 i=

(3.3.30)

Equations (3.3.29) and (3.3.30) are the first twso moments

of a random variable distributed as in (3.3.28). As a

final check on the moments of T, it is known that the

distribution of T is invariant under the transformation

(see [1])

(m,vl,V2) -t (Vl,m,v2+v2-m).. ..1

Simple substitution shows that the moments of T given in

(3.3.21) and (3.3.22) are in fact invariant under this

transformation.

In the null case when X1 = X2 = *** = Am = 0, the

moments of T reduce to

E(TliX -= ** = 0] my' (3.3.32)

-64

and

(V2-m) (v2-m-1) (v2-mn-3) E((T) (13 = *** = Am =

=mv1 (mv2-m2-2m+2) + 2mvl~v2-) 3.3.33)

3.4EauaonoCeanExeatosUd
in Obtaining the Moments of T

In Section 3.3, we listed the values of E -

E d7and E a .-
~1= 11 1

in order to evaluate the moments of T (see equations

(3..1),(3.3.19) and (3.3.20)). Wfe derive these expec-

tations here. Recall that di,***,dm are the latent roots
of S2. Thus

= tr S21', (3.4.1)

dl- = tr IS2-1 2, (3.4.2)
i=1 i
and

1 1 = (tr S2-1 2 tr(S2~']'. (3.4.3)

We shall need some results in matrix algebra and

some w\ell known properties of the Wdishlar~t random matrix

(see [1] or [32]). Let A be an m x m nonsingular matrix

and partition A? into

-65

A21

'----
~22

(3.4.41)

where A21 is a p x p

matrix with p1 + p2 = m.

Then

(3.4.5)

Suppose that B is an m x m matrix satisfying

B = A-'.

Partition B in the same manner as A,

(3.4.6)

(3.4.7)

Then

B1II AII A12 ~22-1 21)-

and

(3.4.8)

B22 122 A21All-1Ax2)-

With A defined and partitioned as above, suppose

that

A Wm(Iv,(0)].

(3.4.9)

Then A1 and A22 are independently distributed and

JA = JAlljA22 A21All- A12 ,

= A22 )11 12 22 121*

B11 i B12

B21 : B22*

- 66 -

q p Iv,0] i = 1,2. (3.4.10)

Also, A.. is distributed independently of [A. -A.A ,..-'A..
w~ith

33 31. i 11 -134 :J

Furthermore, (A.. A..iA..-1A..) is independent of

4..A.. 'A.. ^ Wz (I,p ,(0)). (3.4.12)

It is important to note that the W~ishart distribution w~ith

m = 1 on v degrees of freedom is identical to the X2-dis-

tribution on v degrees of freedom. Thus if

then (3.4.13)

u ^ X 2(0).

With these results, we can proceed to evaluate

E(tr S2 ', E(tr(S2 ')2} and {(Itr S2- ]2 tr(S2-1)2}.
For convenience, we shall work with the corresponding

expectations in terms of A, where A is defined by (3.4.9).

We require E(tr AL) E tr(A L)2 and E{(tr A -~) tr(A-1 2
Let

B = A-'. (3.4.14)

Then, by symmetry,

-67

Eltr A '

=E(tr B)

m

i=1

b..~

(3.4.15)

=mE (b 11).

From equation (3.4.8) with pi = 1 and p2 = m 1,

bii = (aii A12A22-1A21)-

(3.4.16)

Thus using (3.4.11) and (3.4.13)

1
b Xv -m+1OT

13.4.17)

and hence

E(tr A-1) = m .
(v-m-1)'

The evaluation of E{(tr X-')2 tr(A-1)"}

(3.4.18)

is

similar. By symmetry

E((tr A')2 tr(A ') ]

= ((tr B)

tr~(B>)

bij..

E (b Ilbez

m m
C -
=13=1

(3.4.19)

=E ;i m
i1

=m(m-1)

b..1 )

- b 2 )

- 68 -

With pi = 2 and p2 = m 2 and using (3.4.8), wse have

bllb22- b122 = /B1zi

= [A11 Al2A22 121 -1|.

(3.4.20)

Therefore,

(3.4.21)

El(tr A-1 2

Denote (A 1 Al2A 22 A21) by

(All Al2A22-IA21)

=C11

C21

C12

C22 ,

(3.4.22)

so that from (3.4.11)

C W 2II,v-m+2,(0)].

(3.4.23)

Now from equations (3.4.5), (3.4.10) and (3.4.11),

(3.4.24)

w~ith

tr(A-1 2) = m7(m-1) E I.lA1A2

2
IC = cii c22 C12
C11

- 69 -

cil ^ Xy-m+22(0)

and

(3.4.25)

This implies that

E((tr A-')2 ir-1 2 = m(m-1) E 1~ Pi,,12

m(m-1)
= (v -m)(v -m -1)-

(3.4.26)

The evaluation of E~tr(A4- )2) can be carried out

using (3.4.26) and evaluating E(tr A- )2.
metry we have

E(tr A-')2 = E(tr B)2

i=1

11 i 1 3
1=1Ir

Again by sym-

=mE(b112) + m(m-1) E(bizb22).

(3.4.27)

From (3.4.17),

E(bli2) = 1 .

(3.4.28)

To evaluate E(bllb22) COnSider (3.4.7) and (3.4.8) with

pi = 2 and p2 = m 2. We have

C2
C22 1 ^ X2( )
c1lv-+

70-

bll bl2
B11 =
b21 b22

= All Az2A22-1A~!)-1

=C-' (3.4.29)

where C is defined in (3.4.22) and distributed as in

(3.4.23). Another application of (3.4.8) yields

bii = cir i
C22

and (3.4.30)

b22 =22 C122 -1
11

Thus

c1-1
blib22 = 11022 i 1 3..1
C11c22

The remarks preceding (3.4.10) along with (3.4.23) imply

the independence of cll and C22. Also, the remarks pre-
C22
ceding (3.4.11) imply the independence of cll and C22 1
C11
Hence cll is independent of 11 -c'2 )~. Similarly, c22
c11022
is independent of (1 C12 ) and,hence, so is the product
c11C22
cllc22. This implies that

-71

-22
C11 C22 C11022i

= ~ ~ ~ 1 E 2 (3.4.32)
i 11 i 22)"C C122
C111 C2
C22

The expectations E[ and EI can be evaluated using

(3.4.10), (3.4.13) and (3.4.23), yielding

[C111 v-m1

and (3.4.33)

C221 v-m1

To evaluate

C11
[ C22

we need to make use of (3.4.12). Let

c122
u2 = c11
C22

and (3.4.34)
C22
c22

From (3.4.12) and (3.4.23), u, and u2 are independently

distributed as

-72

u' ^ Xv-m,+1 (0)

(3.4.35)

and

u2 ^ X1 (0).

Therefore

C11 1 + U2
C11 C122 U1
22

1
Se(1l,v-m+1) *

(3.4 36)

Equation (3.4.36) yields

(v-m)(v-m-2)

(3.4.37)

Substitution of (3.4.33) and (3.4.37) into (3.4.32) gives

E (b 11b22) = (v -m) (-m- [Tv-m-3), )

(3.4.38)

and wde obtain from (3.4.27), (3.4.28) and (3.4.38)

mm-2-2m+2) .
V m)~v m -1)(v m -3)-

E(tr A-1)2

(3.4.39)

Finally substituting (3.4.39) into (3.4.26) and solving for

E~tr(A ')2):we find that

(v-m)(v-m-1) (v-m-3).

(3.4.40)

/3

Putting A = S2 and replacing v by v2 in equations (3.4.18),

(3.4.26) and (3.4.40) we obtain the equations

E(tr S2-1) = Tm-

(3.4.41)

(3.4.42)

m(v2-1)
(v2-m) (V2-m-1)(v2-m-3)-

E~tr(S2-1) ] =

and

E((tr S2-')Z tr(S2-')) = m(m-1) (3.4.43)

which are the same as equations (3.3.18), (3.3.19) and

(3.3.20).

3.5 Summary

The representation of T given in (3.2.34),

m u.2
i=1 1

can be used to evaluate higher order moments of T. The

expectations involving the elements u1 ,***,um can be
evaluated using (3.3.6) and the orthospherical integrals.

The orthospherical integrals necessary for the first four

moments of T are tabulated at the end of Chapter II. The

expectations involving elements of D can be found using

simple properties of the Wzishart distribution similar to

those in Section 3.4.

.. 7 4

It should be noted that the second nament of T

depends upon

i= 1

This refutes the claim by MIikhail [27] and Srivastava

[33] that the power of T depends upon Al *, only

through

i=1

CH-APTER IV

APPRIOXIMATIONS TO THE DISTRIBUTIION OF
HOTELLING'S GENERALIZED T z

4.1 Introduction

It was pointed out in Chapter III that the

distribution of Hotelling's generalized Tq is well known

in certain special instances. These are the cases when

m = 1 for all vl and V2; when vi = 1 for all m and V2 > m;

and when v2 goes to infinity for all m and vl. In addition

to these special cases, Hotelling [13] derived the null

distribution of T_ for m = 2, all vi and v2 > 2. With

the exceptions of these fewY instances, the distribution

of Hotelling's generalized To is customarily in series

form in terms of the zonal polynomials as described by

James [21] and Constantine [3]. Constantine [4] gives a

series representation of the distribution of T. His series

is convergent, however, only when [IT < 1. In 1968, Davis

[6] showed that the null distribution of T satisfies a

linear homogeneous differential equation of order m, and

that Constantine's series reduces to the solution of this

equation in the case ITI < i. Davis [7] uses this result

to obtain the upper 5% and 1% points of To'/v1 for m = 3,4

-75

- 76 -

and various values of vl and V2 which he claims are

generally accurate to five significant figures.

In 1954, Pillai [29] suggested that the distribution

of T be approximated by

2 1
m (v2-m-1)+2 Fmvz,m(v2-m-1)+2. 411

It should be pointed out that this approximation becomes

exact for m = 1 as well as for the limiting case when v2

tends to infinity~. However, the approximation is not

exact in the case vl = 1, nor is it invariant under the

transformation

10I,V1,V2) 12 ( 1 m~+2- m

In this chapter three approximations to the

distribution of T are considered. Upper 5% and 1% points

are found for each of the approximations in the cases

m = 2,3, and 4 and some accuracy comparisons are made.

4.2 Approximations to the Distribution and
Comparison of Percentage Points of T

In view of the known distribution of T_ for the

special cases mentioned in Section 4.1, a number of approxi-

mations to its distribution are proposed for the general

situation. Of course we would like the approximation to

be a simple one in terms of tabulated functions and to

-77

satisfy as many known results as possible. Naturally, we

would want any approximation to be sufficiently accurate

for practical purposes. For these reasons, the following

three approximations to the distribution of T are considered:

Approximation 1

T I aX l(Y) (4.2.1)

for some al, v, and y;

Approximation 2

T X aF (Y) (4.2.2.)
*mvi,v2-l

for some al and y; and

Approximation 3

T aF (Y) (4.2.3)
.v,v2-m+1

for some al, v, and y.

Approximation 1 was chosen in view of the fact

that the limiting distribution of TO2 iS

X 2 1- *i
mv, .- 1

It was suspected that this approximation would be particu-

larly suited for large v2. AppTOXimation 2 waS chosen

since in the cases m = 1 or vi = 1, the distribution of T

78

is given by (3.3.23) and (3.3.27), respectively. It was

thought that approximation 2 would be fairly accurate

overall. The third approximation was chosen for reasons

similar to, and in hopes of improving, approximation 2.

In approximations 1 and 3, the first tw~o null

moments of the approximating distribution w~ere equated to

the first tw~o null moments of T given in (3.3.32) and

(3.3.33) in order to determine the unknown multipliers,

a, and unknown degrees of freedom, v. In approximation 2,

the first moment of the approximating distribution was

equated to the first null moment of T given in (3.3.32) in

order to determine the unknown multiplier, a. The first

nonnull moment of each of the approximating distributions

was then equated with the first nonnull moment of T given

in (3.3.21) to obtain the unknown noncentrality parameters,

y. Derivration of these unknown multipliers, degrees of

freedom, and noncentrality parameters follow.

Let u be a random variable such that

u aX l(Y). (4.2.4)

The first two null moments of u are known to be

E(uly=0) = av (4.2.5)

and

E(u ly=0) = u'v~v+2).

(4.2.6)

-79

Equating these two expressions w~ith (3.3.32) and (3.3.33)
wLe obtain

(v2 1 vl~2-m
(v2-m (2-m (2-m3

Smv,(v2- m (2-m
() V2-1)(V +v2-m1

(4.2.8)

and

The first nonnull moment of u is given by

E(ulv) = a(v + y).

(4.2.9)

Equating this with (3.3.21) and using (4.2.7) and (4.2.8)

yields

S(v2-1)(Cvl+V2-m-1) i
i=1

m

m i=1

(4.2.10)

where the value of v is given by (4.2.8).

Thus approxi-

mation 1 becomes

T d aX 2(y), (..1

wh~iere a, v), and y are given by (4.2.7), (4.2.8) and

(4.2.10) respectively.

If u is a random variable such that

u amvl,v2-m+1y),

(4.2.12)

- 0

then the first null moment of u is

E~u =0)= a(v2-m+1).
E~ul-~O) = [v2-m-1)'

(4.2.13)

The evaluation of a by equating (4.2.13) with (3.3.32)

yields

C1v2-m+1'

(4.2.14)

Using (3.3.21), (4.2.14) and the first nonnull moment of u,

E(u y) = 0 291 7 2 l
C1mv-(v2-m-1)

(4.2.15)

w\e obtain

m
i = 1
i=1

(4.2.16)

Approximation

2 then becomes

(4.2.17)

m X
T -mT Fmvi,v,-m-1 i .

If u is a random variable with

u av,v,-m+1(),

(4.2.18)

the first two null moments and the first nionnull moment of

u are respectively

E(uly=0) = a v21-m+), (4.2.19)
[v2-m-1)

- 1 -

E(u2ly=0) = 2 (v+2)(v2 m12

( vlyv2--lv-3'

E(u 7) = a (,-~l

(4.1.20)

(4.2.21)

and

A procedure similar to that used for approximation 1 yields

the following values for a, v, and y:

a= mvi ,

(4.2.22)

(4.2.23)

vt = -m
Svl+v2 1l-

and

m
Y V2-m C Xi
vl~v2-mvi-l i=1

i=1

(4.2.24)

It should be pointed out that the value of v given in

(4.2.23) may be negative when m and vl are large, relative

to v2. We W~ill agTSO tO use approximation 3, therefore,

only when the value of v is positive. That is, when

V2 > ml 1+1. Thus, approximation 3 becomes

* mvi v
.2 v2m+lT v ,v2-m+1 mV 1' i~CX1
i=

(4.2 25)

- 2 -

provided v > 0, where

v = v~2O.(4.2.26)
Sv:+v2-mvl--1

Approximations 2 and 3 become exact in the cases

m = 1 for all vl and V2; V1 = 1 for all m and V2; and in

the limit as V2 goes to infinity. They are also invariant

under the transformation

[m,vl,V2) (Vl,m,Vlfv2-m).

Approximation 1 is also invariant under this transformation

and becomes exact in the limit. It is obviously not exact,

however, when m = 1 or vi = 1.

The following tables give the upper 5% and 1%

points of T obtained from each of the three approximations

considered in this chapter, as well as some accuracy com-

parisons. In the case m = 2, our approximate percentage

points are compared with the exact percentage points ob-

tained from the distribution derived by Hotelling [13] as

well as those found using Pillai's approximation [29].

For m = 3 and 4, comparisons are made with Pillai's

approximate percentage points and those given, by Davis (7].

[30]. Percentage points for approximation 1 were obtained

using Percentage ~o-ints; of thie X'2-distribution [28]. The

percentage points for approximations 2 and 3, as well as

vt13 23 33 43 53 63 83 125

E 1.4508 .6807 .4429 .3280 .2603 .2158 .1607 .1064
P 1.3966 .6683 .4385 .3261 .2596 .2156 .1610 .1069
3 1 1.4994 .6861 .4446 .3286 .2606 .2159 .1607 .1064
2 1.4979 .6952 .4524 .3341 .2648 .2192 .1631 .1078
3 1.4473 .6820 .4447 .3297 .2619 .2172 .1619 .1073

E 2.1771 1.0071 .6517 .4812 .3813 .3157 .2348 .1552
P 2.0893 .9849 .6424 .4763 .3784 .3138 .2340 .1550
5 1 2.2284 1.0097 .6515 .4807 .3808 .3153 .2345 .1551
2 2.2945 1.0439 .6703 .4924 .3889 .3212 .2381 .1569
3 2.1741 1.0073 .6525 .4821 .3821 .3164 .2354 .1557

E 2.8802 1.3197 .8508 .6269 .4960 .4103 .3049 .2013
P 2.7658 1.2907 .8384 .6202 .4920 .4076 .3035 .2008
7 1 2.9324 1.3193 .8488 .6253 .4949 .4094 .3043 .2010
2 3.0819 1.3829 .8817 .6454 .5084 4192 .3101 2038
3 2.8828 1.3212 .8521 .6279 .4970 .4111 .3055 .2017

E 4.9409 2.2260 1.4243 1.0451 .8247 .6809 .5046 .3323
P 4.7566 2.1806 1.4050 1.0346 .8182 .6765 .5023 .3314
13 1 4.9920 2.2157 1.4169 1.0402 .8213 .6784 .5032 .3317
2 5.4135 2.3783 1.4998 1.0903 .8549 .7024 .5173 .3383
3 .. 2.2342 1.4288 1.0480 .8268 .6825 5056 .3329

- 3 -

TABLE 4.1

Upper 5% points of T
(m = 2)

V1 13 23 33 43 53 63 83 123

E 2.2704 .9850 .623s9 .4558 .3589 .2959 .2189 .1440
P 2.0541 .9350 .6029 .4445 .3519 .2912 .2165 .1430
3 1 2.1429 .9740 .6068 .4460 .3525 .2914 .2164 .1428
2 2.4103 1.0250 .6438 .4675 .3666 .2820 .2222 .1456
3 2.3025 .9953 .6293 .4593 .3613 .2977 .2201 .1447

E 3.2642 1.3893 .8739 .6361 .4998 .4114 .3039 .1995
P .9624 1.3218 .8456 .6208 .4902 .4050 .3004 .1980
5 1 3.0447 1.3287 .8467 .6209 .4900 .4947 .3001 .1978
2 3.5800 1.4807 .9167 .6609 .5159 .4228 .3104 .2025
3 3.3335 1.4100 .8840 .6422 5038 .4143 .3056 2003

E 4.2220 1.7736 1.1099 .8058 .6320 .5197 .3833 .2513
P 3.445 1.6922 1.0764 .7878 .6208 .5121 .3792 .2495
7 1 3.9124 1.6911 1.0735 .7855 .6191 .5108 .3783 .2491
2 4.7468 1.9223 1.1796 .8461 .6583 .5382 .3939 .2561
3 4.3327 1.8065 1.1258 .8152 .6382 .5241 .3859 .2525

E 7.0198 2.3804 1.7842 1.2879 1.0065 .8255 .6069 .3964
P 6.4298 2.7618 1.7373 1.2634 .9915 .8155 .6015 .3942
13 1 6.4443 2.7339 1.7208 1.2530 .9844 .8104 .5985 .3928
2 8.1696 3.2149 1.9432 1.3809 1.0675 .8687 .6317 .4078
3 .. 2.9509 1. 8182 1.3079 1.0196 .8349 .6123 .3989

- 4

TABLE 4.2

Upper 1% points of T
(m = 2)

V2

v1 30 40 50 60 80 100 200

D 1.0385 .7348 .5682 .4631 .3380 .2661 .1289
P 1.0097 .7201 .5595 .4574 .3351 .2644 .1286
5 1 1.0361 .7329 .5668 .4621 .3374 .2657 .1288
2 1.0936 .7655 .5897 .4768 .3458 .2712 .1303
3 1.0407 .7364 .5695 .4642 .3388 .2667 .1292

D 1.8477 1.3020 1.0045 .8174 .5954 .4682 .2263
P 1.8022 1.2791 .9908 .8084 .5907 .4653 .2257
10 1 1.8379 1.2960 1.0005 .8146 .5938 .4672 .2260
2 2.0026 1.3892 1.0605 .8564 .6175 .4824 .2299
3 1.8579 1.3081 1.0085 .8203 .5972 .4694 .2266

D 2.6327 1.8503 1.4251 1.1584 .8425 .6619 .3193
P 2.5731 1.8206 1.4075 1.1468 .8365 .6583 .3185
15 1 2.6157 1.8402 1.4181 1.1531 .8394 .6599 .3189
2 2.8997 2.0022 1.5234 1.2273 .8819 .6874 .3259
3 1.8616 1.4325 1.1636 .8456 .6639 .3199

- 85 -

TABLE 4.3

Upper 5% points of T
(m = 3)

v1 30 40 50 60 80 100 200

D 1.3504 .9389 .7189 .5824 .4219 .3305 .1589
P 1.2773 .9024 .6973 .5680 .4143 .3261 .1578
5 1 1.3041 .9152 .7047 .5729 .4168 .3275 .1582
2 1.4749 1.0061 .7611 .6112 .4379 .3408 .1614
3 1.3785 .9544 .7288 .5892 .4257 .3332 .1595

D 2.2913 1.5848 1.2101 .9783 .7070 .5534 .2652
P 2.1863 1.5341 1.1805 .9590 .6970 .5474 .2638
10 1 2.1853 1.5442 1.1859 .9623 .6986 .5482 .2640
2 12.6151 1.7620 1.3219 1.0053 7499 5808 2719
32.3590 1.6221 1.2337 .9945 .7161 .5592 .2666

D 3.1977 2.2039 1.6791 1.3555 .9778 .7645 .3654
P 3.0638 2.1406 1.6427 1.3320 .9658 .7572 .3638
15 1 3.0826 2.1469 1.6447 1.3322 .9653 .7568 .3637
2 3.7372 2.5019 1.8684 1.4865 1.0514 .8116 .3773
3 ... 2.2629 1.7163 1.3811 .9920 .7736 .3677

- 6 -

TABLE 4.4

Upper 1% points of T
(m = 3)

V1 30 40 50 60 80 100 200

D 1.3638 .9524 .7313 .5934 .4308 .3381 .1629
P 1.3140 .9274 .7165 .5836 .4257 .3350 .1622
5 1 1.3589 .9491 .7290 .5918 .4299 .3375 .1627
2 1.4621 1.0062 .7653 .6168 .4439 .3465 .1650
31 1.3689 .9556 7335 5951 .4319 .3389 .1631

D 2.4501 1.7053 1.3070 1.0593 .7679 .6022 .2896
P 2.3732 1.6676 1.2847 1.0447 .7603 .5975 .2886
10 1 2.4357 1.6971 1.3017 1.0553 .7654 .6005 .2891
2 2.7213 1.8562 1.4033 1.1261 .8056 .6264 .2957
3 .. 1.7166 1.3143 1.0644 7709 .6041 2901

D 3.5074 2.4358 1.8643 1.5095 1.0930 .8565 .4113
P 3.4069 2.3871 1.8359 1.4910 1.0834 .8507 .4100
15 1 3.4830 2.4212 1.8560 1.5034 1.0896 .8545 .4107
2 3.9696 2.6961 2.0318 1.6266 1.1597 .8996 .4223
3 ... 1.8773 1.5186 1.0982 .8599 .4121

- 7 -

TABLE 4.5

Upper 5% points of T
(m = 4)

V2

V1 30 40 50 60 80 100 200

D 1.7348 1.1892 .9040 .7289 .5253 .4106 .1962
P 1.6178 1.1324 .8706 .7071 .5139 .4035 .1946
5 1 1.6745 1.1595 .8865 .7174 .5193 .4069 .1953
2 1.9498 1.3036 .9749 .7773 .5519 .4273 .2003
31 1.7819 1.2147 .9200 7399 5314 .4144 .1972

D 2.9810 2.0356 1.5441 1.2434 .8946 .6985 .3332
P 2.8126 1.9564 1.4985 1.2139 .8795 .6894 .3312
10 1 2.8785 1.9865 1.5155 1.2244 .8843 .6920 .3316
2 3.5341 2.3341 1.7301 1.3716 .9657 .7437 .3444
3 .. 2.0957 1.5816 1.2690 .9087 .7075 .3354

D 4.1858 2.8501 2.1581 1.7359 1.2471 .9729 .4632
P 3.9706 2.7510 2.1019 1.6999 1.2289 .9619 .4608
15 1 4.0413 2.7801 2.1188 1.7097 1.2335 .9648 .4613
2 5.1018 3.3493 2.4732 1.9531 1.3686 1.0506 .4827
3 . .. 2.2170 1.7760 1.2692 .9869 .4666

-SS

TABLE 4.6

Upper 1% points of T
(m = 4)

- 9

Pillai's approximation, were obtained from Tables of the

Incomplete Beta Distribution [28]. There are five entries

in each cell. For m = 2, these correspond to the exact

percentage points and the approximate percentage points

found from Pillai's approximation and approximations 1,

2 and 3 of this chapter. These are designated by E, P, 1,

2, and 3 respectively. For m = 3 and 4 the exact per-

centage points are replaced by those given by Davis which

are designated by D. Davis' percentage points are then

used as a standard by which to compare approximations.

The omitted percentage points for approximation 3 are

those which occurred in cases where v2 < (m 1) vl + 1.

4.3 Summary

A comparison of the preceding tables reveals the

following generality for all approximations. The 5% points

are uniformly more accurate than the corresponding 1%

points, and the accuracy of both the 5% and 1% points

increased with v2, while decreasing with both vi and m.

Some unexpected results are also revealed. The

5% and 1% points of T computed from the X2 apprOXimatiOR

(approximation 1) are surprisingly close to the exact

values for m = 2 and the values given by Davis in the

cases m = 3 and 4, even for small v!2. In COmnparing this

approximation with Pillai's, the X2 appTOXimation is

- 90 -

generally more accurate for both the 5% anid 1% points.

Some exceptions occur in the 1% points when both vi and

V2 are large.

Although it was hoped that approximation 2 would

be relatively accurate overall, the results were rather

dis appointing This approximnation is, without exception,

less accurate than the others. It is, however, very

simple, and fairly accurate for large values of v2.

The results of approximation 3 were very satisfying.

Both the 5% and 1% points for this approximation were gener-

ally more accurate than Pillai's, and markedly so for small

V2. The decrease in accuracy for increasing vl seems to

be less for approximation 3 than for the other approximations.

The fact that this approximation cannot be applied when

V2 < m- 1) v1 + 1 Should not be too disturbing, since in

most practical situations we would expect v2 to be large,

relative to vi.

Although the comparison of percentage points is

only for the cases m = 2, 3 and 4 and selected values of

vl and v2, it is suspected that the relative accuracy of

the approximations would be maintained for general m, vl

and V2. It is therefore suggested that the distribution

of T be approximated by

,,,i=1

- 91 -

when v2 > (m 1) V1 + 1. In this case the value of v is

given by

my v 2 -m)(432
Sv2-(m-1)vl+1' (4 '2

If the situation should arise when v2 < E 1) 1i i,

the X2 apprOXimation is suggested.

T aX A.(4.3.3)
v mvi 1

in which

a=(v2-1)(vlfv2-m-1 ) (4.3.4)
( V2-m) (v2-m-1)(v2-m-3)

and

y v l2 -m) (v2-m- 3) (4.3.5)
(v2-1)(V2*M2-m-1)