the distribution of the likelihood ratio criterion for...

M.Sc.

THE DISTRIBUTION OF THE LIKELIHOOD

RATIO CRITERION FOR TESTING HYPOTHESES

REGARDING COVARIANCE MATRICES

BY

CHAPUT, LUC

MATHEMATICS

A B S T R ACT

This thesis deals in general with the testing of hypotheses

about covariance matrices, and, more particularly, with

the problem of the distribution of the likelihood ratio

criterion for testing such hypotheses. The author studies

in detail two statistical hypotheses. For the first, he

uses a simple random sample of p-component vectors from

a multivariate normal population N(~,t) to test the 2 2 hypothesis H:t=a l where a is unknown and l is the identi-

ty matrix. For the second, he uses two simple random

samples, one from each of two multivariate normal popula-

tions, to test the hypothesis that the covariance matrices

of these two populations are identical. This thesis, which

started out to be only expository in nature, developed in

su ch a way that the author was able to obtain new results

while studying particular cases of the criteria. Further,

the author discusses simplifications of the existing proofs

and sorne alternate proofs of sorne results.

Short Title: DISTRIBUTION OF TWO MOLTIVARIATE TEST CRITERIA

M.Sc.

LA DISTRIBUTION DU CRlTERE DU

RAPPORT DE VRAISEMBLANCE POUR TESTER

DES HYPOTHESES CONCERNANT DES MATRICES

DE COVARIANCE

PAR

CHAPUT, LUC

MATHEMATIQUE

SOM MAI R E

Il est question dans la présente de tests d'hypothèses concer-

nant des matrices de covariance et, en particulier, concer

nant le prOblème de la distribution du critère de vraisem

blance utilisé comme critère de décision. L'auteur étudie

en détail deux hypothèses statistiques. En premier lieu, il

emploie un échantillon aléatoire simple de vecteurs à p-

composantes tirés d'une population normale multidimension-

( ) ,2 Ù 2 nelle N P,t afin de tester l'hypothese H:t=a l 0 a est

inconnu et l est la matrice-identité. En second lieu, il

emploie deux échantillons aléatoiles simples, un de chacune

de deux populations normales multidimensionnelles, afin de

tester l'hypothèse: les matrices de covariance desdites

populations coincident. La première ébauche de cette thèse

se présentait seulement comme un exposé commentateur mais

par la suite, en analysant des cas particuliers des critères,

l'auteur a réussi à trouver de nouveaux résultats. De plus,

l'auteur considère quelques simplifications de preuves déjà

existantes et quelques autres preuves de certains résultats.

THE DISTRIBUTION OF THE LlKELIHOOD

RATIO CRITERION FOR TESTING HYPOTHESES

REGARDING COVARIANCE MATRICES

BY

CHAPUT, LUC.

THESIS SUBMITTED TO THE

FACULTY OF GRADUATE STUDIES AND RESEARCH

IN PARTIAL FULFILMENT OF THE

REQUlREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE.

DEPARTMENT OF MATHEMATICS

MC GILL UNIVERSITY

MONTREAL, QUEBEC

CANADA. DECEMBER 1969

1 0 Chaput, Luc 1970

***************************************

ACKNOWLEDGEMENT

My most sincere appreciation and thanks

to

Professor A. M. Mathai

for his undeniable guidance and support

in the writing of this thesis.

***************************************

CHAPTER 1

CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

CHAPTER 6

CHAPTER 7

CHAPTER 8

CHAPTER 9

CHAPTER 10

BIBLIOGRAPHY

TABLE OF CONTENTS

PRELIMINARY RESULTS

A STATEMENT OF THE PROBLEM

THE CRITERIA

THE MOMENTS OF THE CRITERIA

THE DISTRIBUTION OF THE CRITERIA

ASYMPTOTIC EXPANSIONS OF THE

DISTRIBUTIONS OF THE CRITERIA

PARTICULAR CASES OF THE CRITERIA

THE EXACT DISTRIBUTION FOR THE

SPHERICITY TEST IN THE MOST

GENERAL CASE

APPLICATIONS

CONCLUSION

pages

1 - 9

10-12

13-16

17-21

22-24

25-27

28-42

43-62

63-77

78-79

80-92

C~P~Rl

PRELIMINARY RESULTS

In this chapter we state some useful results and definitions

which will be used as we progress in the thesis.

(1) The univariate normal density function is given by 2 2

(2)

f(x) = l e-(=-~) /20 ~ _m<x<m~_œ<~<m~o>O.

'2~ 0

Let x ~x ~ ••• ~x be a sample of size n from a population 1 2 n

with density f(x;e ~e ~ ••• ~ek) where n~ the parameter 1 2

space~ is the totality of aIl the points that (el~e2~ ••• ~

Bk) can assume. On the basis of this sample~ suppose that

it is desired to test the hypothesis Ho~ (el~e2~ ••• ~ek) is

a point in w. The alternative hypothesis is _Ha~ (el~e2~

••• ~ek) is a point in n-w';. The likelihood of the sample n

is L=.n f(x.;el~e2~ ••• ~ek) where L is a function of the ~=l ~

parameters and will usually have a maximum (usually found

by differentiation) as the parameters are allowed to vary

over the entire parameter space n; we shall denote this

maximum value by L(fi). In the subspace w~ L will also

ordinarily have a maximum valua denoted by L(w).

(3) The likelihood ratio criterion is defined by

À=L(w)/L(n) where L(w) and L(n) were defined in (2).

(4) The observed sample variance defined by ~ (x._x)2 / (n_l) i=l ~

2 n - 2 is a value assumed by the random variable S = L (X.-X) + i-l ~

(n-l) at the point (xl~ ••• ~xn) of the sample space.

2

(5) The F distribution with m and n degrees of freedom has

a density function defined by

f(x)= r(m~n)/2 (m/n)m/2 xm/ 2- 1/(1+mx/n)(m+nJ/2 r(m/2)r(n/2)

where x>O, m and n are positive integers and

r(t) J:x t - 1 e-:: dx for t>O.

(6) The p-variate normal density function is given by

where x' = (xl ,x2' ••• ,xp ) is a random vector, EX=l1 and

t=E(X-11) (X-11)' for t positive definite.

(7) If xI, ••• ,xN constitute a sample from a p-variate normal

population N(l1,t), the maximum likelihood estimates of 11

and t are ~=x=; xa/N and

,. t=;{xa-x)(xa-x)'/N=A/N.

(8) Let T be a random vector with density g(r,s), Sen.

(9)

Let Hl be 8en lcn, H2 be een 2c n l , given Sen l , let H be

Sen 2 , given Sen. If Xl is the likelihood ratio criterion

for testing HI ,À 2 for H2 and X for H are uniquely deter

mined for the observation vector T, then À=X I À2 •

If XI, ••• ,X have a joint normal distribution, a necesp

sary and sufficient condition that one subset of the

random variables and the subset consisting of the

3

remaining variables be independent is that each covari

ance of a variable from one set and a variable from the

other set be O.

(la) When t is diagonal, the correlation coefficients[rii)

are independently distributed of the variances

(au/(N-1') . (11) If the p-component vectors x 1 ,x2 , ••• ,xN (N)P) are inde

pendent, each with the p-variate normal distribution, N

then the density of A = 1: 1 (x -x)(x -x)' is given by_ a:::: a a

lAI (n-p-l)/2 exp-~tr A1:-l/2npI2 wP(P-l)/4ItlnI2

.. ~ r [(n+l-i) 12 ] 1,==-1

where n=N-l; it will be denoted by w(A/1:,n).

(12) K(t o ,n)/K(to ,n+2h) == 2hp

i!1 aiih i~lr [(n+l-i)2+h] /

.r [(n+l-i)/2.]

(13) If the dispersion matrix A is w(A/t,n) distributed,

then a la has the Chi-Square distribution with n ii ii

degrees of freedom, defined by (14).

(14) The Chi-Square distribution with n degrees of freedom

( 2) Xn

has a density function defined by

nl2-l x -%/2

e

where n is a positive integer and x>O.

(15) The hth moment of the random variable having a Chi

Square distribution with 2p degrees of freedom is

2h r(p+h)/r(p).

(16) If x., i=1,2, ••• ,n are independent random variables 1.

having the

degrees of

Chi-Square distribution with v1 ,v2 , ••• ,vn n

freedom respective1y, then .t x. has the 1.=1 1-

4

Chi-Square distribution with v-tvi degrees of freedom.

(11) K(t,ng)/K(t,ng+ngh) =

2,nghPltllngh i~ï r[(ng+ngh+1-i)/2]/r[(ng+1-i)/2]

(18) If ~he Ai (i=1,2, ••• ,q) are independent1y distributed

according to w(t,ni) respective1y, then A=tAi is distri

buted according to w(t,tni).

(19) The fo11owing is known as Gauss' Multiplication Formula

where n is a positive integer and Z is such that the

Gammas are defined.

(20) The Mellin Transform of the density f(x), x>O, is given

by

~ (s) -;: j: xs - 1 f(x)dx = E(X)s-l

The Inverse Mellin Transform of the (s_l)th moment of X

about the origin is

jC-l-i CD

f(x) = (1/21fi) ~(s) C-iCD

(21) mn Gpq

(1/21r1)

al'·· · ,ap l bl,···,bqJ

Ir m n n r(bo-s) n r(l-a;).+s) x8 ds

j=l ;) j=l L __ _

~ r(l-b.+s) ~ r(a.-s) j=m+l ;) j=n+l ;)

5

where 1= {-l and an empty product 1s 1nterpreted as l,

O~m~q, O~~, and the pe~ameters are su ch that no poles

of r(b.-s), j=l, ••• ,m co1nc1des w1th any poles of ;)

r(l-ak+s), k=l, ••• ,n. There are three d1fferent paths

L of 1ntegrat1on, one 15:

L runs from -1œ to +1œ 50 that aIl poles of r(b.-s), ;)

j=l, ••• ,m are to the r1ght and aIl the poles of

r(l-ak+s), k=l, ••• ,n to the left of L.

(22) ~

where 1:{-l,

h(s) = ~ r(l-a.-a.s) W r(b.+B.s) j=l ;);) j=l ;);)

q p n r(l-b .-13 .5) . n r(a;)o+a;).s)

j=m+l ;);) ;)=n+l

and O=n~p, l~m~q, aj' jcl,2, ••• ,p and Bj' j:l, •• ~,q

are positive real numbers and aj' bj may be complexe

Further, aj(bh+v) #= Bh(aj-l-À) for v,À,= 0,1, ••• , and

h=1,2, ••• ,m; j=1,2, ••• ,n. The contour L 15 su ch that

the set of points s=(-b.-v)/B., j=1,2, ••• ,m; v=0,1,2, ••• , ;) ;)

and s=(l+v-aj)/aj' j=l, ••• ,n; v=0,1,2, ••• , are separated.

(23) Consider a random variable W(O<W<l) with hth moment

defined by

E(W)h = ~ j[l y / j / k~l xk "'k) h kU1 r [Xk (Hh)+~kJ •

b •. II r[y.(l+h).,.n.] 3=1 3 3

o a b

6

where K is such that EW =1 and k~lxk = j~lYj and xk ' t k ,

y., n. are such that there is a distribution with such 3 3

moments. Let M= -2logW and O~p<l. Then the cdf of .

-2plogW is given by

Pr(M~Mo) = Pr(pM~pMo)

0= Pr (X;. 0 Mo) + ~ l ( Pr ( X; J 0 Mo)

+ [~2 ( pr( X;~rMo) - Pr (X; .. OMo))

-2Pr( X;S OMo) + pr( X/OMo) ) J + ••• +R:t1

T Rv ( - (m+1) ) d he error, m+1 is 0 e if xk~cke, Yj} je,

ck>o, dj>O and (l-p)xk , (l-p)Yj have limits, where p

May depend on e. In Many cases, it is desirable to

choose p such ~hat 001=0; in such a case using only the

-2 first term of the expansion gives an error of order e .

Also, f== -2 [Et i - En j - l(a-b)]

~,,= [(_1)"01 Ir(r+1)] [~B"+l ($k+~k)1 (OXk)" -

~B"'+l (e: .+n ')/(Py.)Z"] ;) ~- 3;) 3

7

polynomial of degree rand order unit y defined by

(24) The univariate Beta density function is given by

1.' a,S>O and further EX ;::ll;::: r(a+s)r(a+r)/r(a)r(a+s+r)

-1 and r(a+S)/r(a)r(s)~B (a,S).

(25) If Re c>Re b>O, the following is known as Euler's formula

F(a;b;c;z) =,[r(C)/r(b)r(C-b~ J~ tb-1(1_t),,-b-l (l-tz)-a dt

where Izl<l and Re( ) denotes the real part of ( ).

(26) CIO

F(a;b;c;z) = t (a)n (b)n znln!(c)n where c is neither n=O

zero nor a negative integer,

(a)n = (a)(a'f'1)(a+2) ••• (a+n-l) = r(a+n)/r(a) for n~l

and ( a ) 0 ::: 1, a~ 0, 1 z 1 < l •

c-a-b ( (27) F(a;b;c;z) _ (l-z) F c-a; c-b; C; z) is valid if

Izl<l, and a, b, c-a, c-b are non-negative integers.

(28) A corollary to the Monotone Convergence Theorem is as

follows: Let un be a sequence of non-negative measura

ble funct10ns and let

8

(29) The cdf F(x) of the Beta distribution Be(a,B) designated

by l (a,B) and called the incomplete Beta function has :r:

been tabulated under the direction of Karl Pearson

for x=O.Ol to 1.00 and for a,B~0.05 to 50.

(30) This is a result obtained by Consul [16]. For O<x<l,

the Inverse Mellin Transform

-1 jC+iCO ••

(2ni) x-8 r(ps+a)r(ps+b)ds, i=/-l, = c-ico r(ps+a+m)r(ps+b+n)

xa/ p (l_xl/p)m+n-l F(n;a+m-b; m+n; l_x1/ p ) p r(mof-n)

(31) Consul also obtained for O<x<l and of-i=/-l

-1 r+ iCO (2ni) x- 8 r(qs+a)r(qs+b)r(rs+c)ds _

c-ico r(qsof-a+m)r(qs+b+n)r(rs+c+p)

.. t (n). (a.m-b). [j! (m_n_j _l)]-l (l_xl/Q)i J=O J J

.F ( l;l-a+(c+i)q/r; m+n+j+l; _(1_x1/ Q) x-1/ Q)

(32) The follow1ng 1s known as Euler's constant y

y=11m (Hn-logn), where Hn = ~ l/k; it 1s known that n+co k=l

y=0.5772, approx1mately.

(33) The ~ functicn 1s defined as co

~(a) = E (a-l)/(m+l) (a+m), a#O,-1,-2, ••• , and exclud1ng m=O

Euler's constant y.

9

(34) The genera11zed Zeta function 1s def1ned as

• g(s,v) = m;O 1/(v+m)8, Re(s»O, vFO,-1,-2, ••• , where

Re( ) denotes the real part of ( ).

10

CHAPTER 2

A STATEMENT OF THE PROBLEM

In [~ Anderson discusses the likelihood ratio criteria for

several statistical hypotheses; here we will consider two

of these hypotheses. For the first, we use a simple random

sample of p-component vectors x1,x

2, ••• ,x

N trom a multivariate

normal population N(p,t) to test the hypothesis H,I=a21 where

a2 is unknown and l is the identity matrix. For the second,

we use a set of simple random samples, one from each of two

multivariate normal populations, to test the hypothesis

Hl,Il-I2' that is the covariance matrices of these two pupula

tions are identical. Anderson derives the likelihood ratio

criteria for these two hypotheses, finds their moments under

the null hypothesis and studies a few particular cases.

The purpose of this thesis is to write an expository article

on the recent results and give a complete coverage of the work

done on these two problems. The results which are available

in books will not be discussed in detail. The recent works

on these problems will be summarized. Further, simplifica

tions of the existing proofs and some alternate proofs of some

results are also discussed.

We will show how to derive the likelihood ratio criteria for

these two hypotheses; we will mention modifications when

evaluating the moments of the criteria; we will derive the

Il

density or the criteria; we will look at asymptotic expan

sions or the distributions or the criteria; we will discuss

new results while looking at particular cases or the criteria

and we will give some applications.

To get acquainted with the second hypothesis, HI,tl~t2'

consider it's univariate analogue: We are given two 2 2 univariate normal populations N(~l,ol) and n(~2,o2)' and

2 2 we would like to test the hypothesis HI ,ol=o2 against the 2 2 alternative hypothesis Ha ,ol*o2.

By (3), (2) and (1) the likelihood ratio criterion is given

by

(2.01) À" (m+n) 12,,[ t (xl i-Xl ). +t (X.ri • ).]) (m+n) /2

[ / ( - )2] -m/2 [/ ( - )2] -n/2 . m 2~t Xli-Xl n 2~t X2j-X2

where m and n are the sample sizes. This criterion

may be written in the rorm

(2.02) À ~m+n)(m+n)/~mm/2 nn/j .(~:i F)m/;!lIT~:i Fl (m+n)/2

where F is the variance ratio (4) derined by

which has the F distribution with m-l and n-l degrees

or rreedom (5) when Hl is true.

12

On p10tting A as a function of F, it is apparent that

the critica1 region O<A<A corresponds to a two-tai1ed

test on F.

2 We will notice that the first hypothesis H,t=a l has no

univariate analogue.

CHAPTER 3

THE CRITERIA

13

We will derive first the likelihood ratio criterion for

testing the hypothesis Hl' t l=t 2 • Let X~ (a-l, ••• ,N ;g=1,2) g

be an observation from the gth p-variate normal population

N(pg,L g ) defined by (6).

(3.01)

(3.02)

(3.03)

(3.04)

(3.05)

2 Ng g 2 Let N = t Ng , Ag= t

1 (x~-xg) (xa-i

g) " A== t Ag.

g:=l a= g::l

By (2), the likelihood function is found to be

where the space n is the parameter space in which each

t is positive definite and pg any vector and the g

space w is the parameter space in which tl

==t2

and

pg any vector. The function to be maximized in w is

By (7), 'O~ == xg and f w:= AIN where A and N are defined

in (3.0U; (3.03) becomes

In a similar fashion, we obtain tgn AgINg

so that (3.02) becomes

14

By (3), the 1ike1ihood ratio criterion is the quotient

of (3.04) and (3.05) so that we get

(3.06)

(3.07)

For the work that fo110ws, we will use the definition

9f Bart1ett [7] ; the statistic he proposes is defined

by

~ .J

V lA 12ng

1= g g=l

For the case p=l, the critical region Vl~Vl(a) is based on

the F-statistic with n l and n 2 degrees of freedom, as we

already know from chapter 2. Brown ~q and Scheffé ~~

have shown that F~Fl(a), F~F2(a) yields an unbiased test.

To derive the 1ikelihood ratio criterion for testing H, 2 E=o l, we have modified Anderson's method so as not to

consider H as a special case of testing independence of sets

of variates. We use the fact that H is a combination of

two hypotheses: Hl' the components of X are independent

and H2' the variances of the components of X are equa1 given

the components are independent. By (8), the likelihood

ratio criterion À for testing H is the product of the like1i

hood ratio criterion Àl for Hl and À2 for H2• By (9), Hl is

equivalent to the equation E:Eo=(oij Ôij) where Ôij is the

Kronecker Delta which is equal to one if i=j and zero if

i:f:j.

15

By (2) and (6), the 1ike1ihood function is given as

(3.08) L(a)=œ~1{2~)-IP Itl-I exp -i{xœ-~)'t-l(xœ-~)

(3.10)

(3.11)

where a is the space in which t=(a ij ), i,j=1,2, ••• ,p.

By (7), the maximum value of L(a) becomes

i,j=1,2, ••• ,p. Simi1ar1y we obta1n

L(Q)= (2~)-IPN IÉol-IN exp -ipN, where fo={a .. ô • • )/N, 1,;) 1,;)

so that Àl 1s eva1uated as

À 1 = 1 A 1 lN ( ~ a .. rlN i=l 1,1,)

Proceed1ng in the same fashion, we eva1uate À2 as

(3. 12) A2 = ('&lau) IN ( ~ i~laiilPN 12• so that

(3.13) A= IA1 NI2 (~ E aiijPNI 2

2 We notice that the hypothesis H, I=a l can be put in

the form that aIl the roots of

(3.14) II-~II=o are equal, or that the arithmetic Mean of

(3.15 )

the roots ~1'~2""'~P is equal to the geometric

Mean, that is

Since the

squares of the lengths of principal axes of e1lip-

soids of constant density are proport10na1 to the

roots ~i' wh1ch are now equal, the hypothesis H

imp11es that the ellipsoids are spheres.

16

Mauch1y [43] defined a significance test for finding the e11ip

ticity in a harmonie dial. In a subsequent paper [44], he

modified his test to define a criterion for determining the

sphericity of a normal p-variate distribution and also

obtained its moments under the null hypothesis. Girshick [27]

obtained the distribution of the ellipticity statistic under

some special conditions. Hickman [30] has given an examp1e

for obtaining the confidence regions for the dispersion

matrix if it is taken to be proportional to any given matrix.

Ilun [34] has discussed a number of such criteria in the case

of multivariate normal distributions.

17

CHAPTER 4

THE MOMENTS OF THE CRITERIA

To derive the moments of the 1ike1ihood ratio criterion for 2 testing the hypothesis H, E=a l, we have made modifications

so as not to consider independence of sets of variates, as

Anderson [lJ. Slnce IAI=lrijlnaii' Àl depends on1y on

(r i,i) and À2 depends on1y on (aH) ; using (10), we assert

that Àl and À2

are independent1y distributed when H is true,

and, therefore,

(4.01)

(4.02) Let W __ À2/N, W 2/N W 2/N th W W d W

l="Àl '2=À2 en, l' an 2

are monotonie increasing function of À, Àl' and À2

respective1y. By (11), we get the hth moment of

Wl as

h J fi h -h (4.03) EW 1= ... lAI naii w(A/Eo,n)dA, where

Eo=(a .. ô •• ). By a simple a1gebraic manipulation 1,.;) 1,.;)

(4.03) becomes

(4.04) Ew7=~(to ,n}/K(t o ,n+2h~ J .. fnaH -h w(Alt o ,n+2h}dA,

where K(E o ,n)/K(E o ,n+2h) is defined by (12).

-h But (4.04) may be regarded as E(naii ) where aii 1s .

the diagonal e1ement of A having a w(E o ,n+2h)

distribution.

(4.05)

(4.06)

(4.07)

(4.08)

18

We obtain

h EW I =

[K(ro.n)/K(ro.n+2h~ E ( p -h) II a.. • i=l 1.1.

By (13), (4.05) can be written as

EW h _

[K(; o-.n)/K(~o .n+2h~ E [ i~/ii -h rX!+2~ -h] Since the (a ii ) are independent1y distributed under

the hypothesis Hl' the components of X are independ

ent, we obtain

[ -h (2 ) -h] E rii Xn+2h

By (15) and (12) and upon simplification, we get

h EW I =

[rp(n/2)/rP(n/2+h~ i!l [r(n+1-1)/2+h] Ir [(n+1-1)/2J

We also have

(4.09) W2 =

ppua . .1 U:a 0 .)p which can be written as 1.1. 1.1.

(4.10) W2 =

pPIIb 0 .1 (Eb .. ) P where b .. =a 0 01 a2 has the Chi-Square 1.1. 1.1. 1.1. 1.1.

distribution with n degrees of freedom.

(4.11)

(4.12 )

(4.13)

(4.14)

(4.15 )

19

h EW 2 = pph f .. which can be written as

J f -ph h - Ebii/2 n/2-1 • • • ( E b .. ) (!lb •. ) e nb • . db Il. • • db pp ~~ ~~ ~~

where KI=1/rP(n/2)2np/2. (4.12) can be written as

h EW 2 =

ph f j -ph -Eb· ,/2 n/2+h-1 p KI IK 2 •• • (Eb ii) K2e ~~ nb ii db 11 ••• db pp

where K2=1/rP(n/2+h)2np/2+ph.

But (4.13) may be regarded as E(Eb .. )-ph where b .• ~~ ~~

has a Chi-Square distribution with n/2+h degrees of

freedom.

By (16) and (15), we get

h EW 2 =

pphr (np/2)r P(n/2+h)/r(np/2+Ph)r P(n/2)

so that we fina11y obtain

h EW =

rPhrCnp/2l/rCnp/2+Phl] .I/ [Cn+l-il/2+h]Ir [Cn+l-1)/2]

To derive the h th moment of VI defined in (3.07), we

again modify Anderson's proof which is done by consid

ering the joint moment of VI and V2 where V2 is a

(4.16)

(4.17)

(4.18)

(4.19)

20

monotonie increasing function of A2' the 1ike1ihood

ratio criterion for testing the equa1ity of mean

vectors given the covariance matrices are identica1.

By (11), (4.16) can be written as

where K(t,ng)/K(t,ng+ngh) is defined by (17).

If we consider the transformation A=~Ag' then

by (18), (4.17) can be written as

By (11), this reduces to

h EVl = ~L K(E ,ng)/K( E ,ngTngh)] K(E ,n+hrj!K(E ,n)

where K(t,n+hn)/K(t,n) is the inverse of (17) with

ng rep1aced by n. Upon simplification, we get

(4.20)

(4.21)

i~l (g!l r [(ng+hng+l-1. l 12] / r [<ng+l-lll2] )

.r [(n~1-i)/2] Ir [(n4-hn.pl-i) 12]

If P is even, say p:2r, we can use (19) and we

obtain

EVh = 1

21

.~ [ ~ rCng+ngh+1-2j)/rCng+1-2j)] r(n+1-2j)/r(n~hn+1-2j) :J =1 g=l

On the basis of these moments, we can express VI as

a product of variates Xa(l_X)b where the X's are

independent1y distributed with Beta densities.

Wilks [83J has given some other integral representa

tions. Rogers [70] has obtained the density of a

variable of the form Xa(1_X)b described above.

22

CHAPTER 5

THE DISTRIBUTION OF THE CRITERIA

We will now use a rather elegant result round by Mathai and

Saxe na [40]. Let W be der~ned by (4.02), then the (s_l)th

moment of W is given as

(5.01) ~~ WS-1f(w)dw= [p(B-IJPr(np/2)/r(nP/2+SP_P~

. i!l r [ (n+l-i) /2+S-1] Ir [(n+l-i) /21

Applying (19) on r{np/2+sp-p), (5.0l) becomes

(5.02) ~ WS-1f(w)dw = [r(npI2H2T) (P-V/2/p(pn-V/2 JLr [(D+1-i) 12]]

.i~l r[(n+l-i)/2+S-1] /r[{n/2+(i-l)/P+S-l]

By taking the Inverse Mellin Transrorm derined by (20)

of the (s-l)th moment or W about the origin, we get

(5.03) f(w) = [r (np/2) (2T) (p-l J/2/p (pn-V /2 i~/ [(n+1-i)l2] ]

jc 'f-i CIO p .1/2ni W-8.~ r[{n+l-i)/2+s-1]/r[n/2+(i-l)/P+S-l]dS

c-iClO 1. 1

Let S = -s, then (5. 03) becomes

where a j =n/2t(j-l)/p-l, j~1,2, ••• ,p

b j :(n-l-j)/2, j:l,2, ••• ,p

pO [ 1 al, • •• ,a ] and Gpp w P is defined bI, ••• ,b

p

We notice that f(w) i5 uniquely

the range of W is finite ~~ •

in (21).

determined since

Now, let Vl be defined as in (3.07); we already

evaluated the hth moment of Vl as

(5.05) EV7= i~l( gLr [(ng+hng+l-i) l2l!r [(ng+l-i) /2]) • r [ (n+l-i) 12] 1 r [(n+hn+l-i) 121

This is the same as

(5.06) EV~:::i~l r[ (n+l-i)/2] Ir [(nl+l-i)/2] r[(n2+l-i)/2]

23

• i!lr[(nl1'hnl+l-i)/2] r [(n2+hn2+l-i)/2] Ir [(n+hn+l-i)/2]

Taking the Inverse Mellin Transform of the (S_l)th

moment of Vl about the origin, we get

(5.07) f(vl)=constant.

l/2~iJc+iœVl-8.~ r (n2s+l-i)/2 ds c-iœ ~=l~r~~~~~~~~~---------

Let S=-s, then (5.07) becomes

24

Using (22), (5.08) can be written as

CHAPTER 6

ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS

OF THE CRITERIA

th n/2 From (4.15) the r moment of W = Z is given by

This is the form (23), with a=p, b=l, xk=n/2,

YI=np/2, nl=O, ~k=(l-k)/2 for k~l, ••• ,p. Thus,

the expansion of (23) is valid with f=p(p+l)/2-l.

To make the second term in the expansion zero, we

take p su ch that

p=2p2-.p-.2/6pn 3 2 2 2 Then, ~2=(p-.2)(p-l)(p-2)(2p +6p +3p+2)/288n p •

25

Thus, the cumulative density function of W is given

by

(6.02) Pr (-2p1ogZ~z)=Pr(-nplogW~z)=

We again make use of (23) to obtain an asymptotic

expansiop- of the distribution of VI defined by (2.07).

The expansion is in terms of n increasing with k l ,

k2 fixed, where we assume ng=nkg' k 1+k2=l.

The hth moment of W1 defined by

(6.03)

(6.04)

26

w == rnnp/2 ~ n -"png/ 2] V 1 L g=1 !T 1

is given by

h [P n/ f 2 P n/2]h EW1=K .n (n/2) n.n (n 12) g ;1"-1 g= 1 1.=1 g

. ~ h r rn (1+h)/2+(1-i)/2'/J r[n(1+h)/2+(1-j )/2' g= 1 i=1 L g ] J :: 1 J

This is of the form (23) with b=p, Yj=n/2, nj=(1-j)/2

for j=1,2, ••• ,p, a=2p, x~ng/2 for k=(g-l)p+l, ••• ,gp,

and g=l,2, tk=(1-i)/2, k:i, p+i and i=1,2, ••• ,p.

Then f=-2 [ttk-tn j -(a-b)/2]=P(P+l)/2 and Ej=n(1-p)/2

and Sk=ng(1-p)/2. In order to make the second term

in the expansion vanish, we take p as

p=l-(tl/n -1/n)(2p2+3p-l)/12(p+l) g

[ 2 2 ~ 2

Tben (a)2=P(pi-l) (p-I)(p+2) (tl/ng-I/n )-12(I-p) J'48 p

Thus, we obtain

(6.05) Pr(-2PIOgWl~F=pr(X~Z)+~2[Pr(X~+J'Z)

- pr( X i'Z)] +O(n -.)

We note that Box [9] has considered W1 in sorne detail.

The general expression for the moments of the likelihood 2 statistic had been used in certain cases to obtain a x

2 approximation and an asymptotic X series for the dlstribu-

tion of the logarithmic statistic M, a modified form of the

logarithmic statistic used by Neyman and Pearson. Box

investigated the method for two genera1 criteria:

1) The test of constancy of variance and covariance of k

sets of p-variate samp1es.

27

2) Wilks' test for the independence of k sets of residuals,

the Ith set having Pz variates.

Box obtained in each case:

1) A series solution which agrees very c10sely with the

exact distributions.

2) An approximate solution using a single x2 distribution.

3) A rather better approximation using a single F distribu

tion.

28

CHAPTER 7

PARTICULAR CASES OF THE CRITERIA

In chapter 4, we used a simple random sample of p-component

vectors x1, ••• ,xN from a multivariate normal population 2 2 N(U,t) to study the hypothesis H, t=a l where a is unknown

and l is the identity matrix. Let W=A 2/ N where A is the

1ike1ihood ratio criterion for testing H. The hth moment

of W is eva1uated as

(7 • 01) EWh = pph r (np /2 )

r(np/2+ph)

where n=N-l.

~ r (n+1-i)/2+h i=l

r [(n+1-1)/2

The distribution of W for p=2 is obtained in [ 1 ].

Consul [17] has given a method, based upon the

Inverse Mellin Transform and Operationa1 Ca1cu1us,

to obtain the distribution of W for p=2,3,4, and 6.

If we consider a set ~f p independent Beta variates

(24), X1 ,X2 , ••• ,Xp and W=X 1X2 ••• Xp then

Now if Xi is Beta distributed with parameters ai,Si'

then E(X1X2 ••• Xp )h coincides with the hth moment in

(7.01). Further, since O<W<l, a moment sequence will

29

uniquely determine the density of W. Hence W can

be considered to be a product of p independent Beta

variates as mentioned above. This is a known fact

and we will use this and apply simple algebraic

methods to obtain the density of W in some particular

cases. This alternate method which is discussed below

is simpler and further the author has not seen it

discussed in the literature; hence, the author

assumes that the method is new.

Case 1. For p=2, by the use of (19), we get

(7.02) Let s=2h, then we obtain

{7.03) E(W1/ 2 )8=r(n)r(n-l+s). Thus, if we identify with a r(n+s)r(n-l)

Beta variate, we see that W1/ 2 has a Beta distribution

with parameters n-l, 1.

Case II. For p~3, by the use of (19), we get

(7.04) EWh_r(n/2-l+h)r(n/2-l/2+h)r(n/2~1/3)r(n/2+2/3) r(nI2-1/2)r(nI2-l)r(nI2~113+h)r(nI2+2/3~h)

Thus W is the product of two independent Beta variates,

the first with parameters n/2-l/2, 5/6 and the second

with parameters n/2-l, 5/3.

(7.06)

30

Let w=uv, U and V independent with U a Beta variate

with parameters a1,Bl and V a Beta variate with

parameters a2 ,B 2• Let f(u,v) denote the joint

density of U and V, that is

where K = r(al+~1)r(a2+B2) r(a1)r(a2 )r(B1)r(B2 )

Consider the transformation w~uv and w2=v. The joint

density g(w,w2 ) of W and W2 is given as

-a +a -B B -1 a -1 B -1 g(w,w2 ) = K(w2 ) l 2 l(l-w2 ) 2 w l (w2-w) l

Hence the density of W is obtained as

(7.07) h(w) =

KWal-~W -al+a2-Bl(l_w )B 2-1(w _w)B l -1dw 2 2 2 2 W

Let w2=(w-l)t+l, then (7.07) becomes

(7.08) h(w)=

KWBl-l(l_W)Sl.S'-~tS,-l(l_t)Sl-l [l-(l-w)~ -Bl+B,-Sldt

using result (25), we get

(7.09) h(w)==

r(a 1+B 1)r(a2+B 2) wal-1(1_w)Bl+B2-1F(a;b;c;z)

r(al)r(a2)r(B 1TB 2)

where a=Cl 1-Cl2 +8 1 , b=8 2 , c=8 1 .8 2 , z=-l-w and

F(a;b;c;z) is Gauss' hypergeometric series (26).

Substituting the actual values or Cl 1 ,Cl2 ,8 1 ,8 2 in

(1.09), we obtain

(1.10) h(w)=

which agr~es with Consul ~~ , arter using (21).

31

Case III. For p=4, by the use of (19), we rind that

W1

/2 is distributed as RT where Rand Tare independ

ent1y distributed s(n-l,l) and s(n-3,112) respective1y.

Using case II the density of M=RT is given as

(1.11) f(m)=

r(n)r(n.1/2) mn- 4 (1-m)?/2F(3/2;1;9/2;1-m) r(n-1)r(n-3)r(9/2)

1/2 Let w = m then the density of W is obtained as

(1.12) h(w)=

which agrees with Consul ~~ •

(7.14)

32

Case IV. For p~5, W is distributed as a product of

four Beta variates Xl' X2 , X3, X4 with parameters

(n/2-1/2, 7/10), (n/2-1, 7/5), (n/2-3/2, 21/10),

(n/2-2, 14/5) respective1y. To find the density of

XI X2X3X4 , consider the transformation u2=u l x 3 and

s=u I where UI= xl X2 • The joint density of U 1 and

X3 is given as

where K=r(al+8 1)r(a2+82)r(a3+83)

r(al)r(a2)r(a3)r(83)r(81+82)

and 8 .8 =21/10 is neither zero nor a negative integer. 1 2

Rence the density of U2 is obtained as

a -1 f (U2 )=KU2 3 •

JI sal-a3-83(1-S)Bl+B2-1(s-U2)83-1F(al-a2+Bl;82;Bl+B2;1-S)ds u

2

Let s=(u2-1)t+1, then (7.14) becomes

(7.15) f(U2)=KU2a3-1(1_U2)BI+82+S3-1.

JI 8 +8 -1 8 -1 a -a3-S3 ot 1 2 (l-t) 3 (l-yt) 1 F(al-a2+81;S2;BI+B2;yt)dt

where y=1-u2 • By the Monotone Convergence Theorem

(28), (7.15) can be written as

(7.16 )

(7.17)

33

c -1 8 +8 +8 -1 - k f(U2) ::=. KU2 3 (1-U2) 1 2 3 t (y )k(1-U2) • k=O

~~t8,~82~k-l(1_t)83-1[1_(1_U2)~ u,-u3-8 3 dt

where (Y)k=(c1-c2+8 1)k(S2)k

(Sl+S2)k k!

Using (25), we get

!1-U2 1<1,

C=Sl+S2-S3.k is neither zero nor a negative intege~

and where

(C)~(CI-C2+S1)k(82)k

(8 1+8 2+S 3)k k!

c=r(Cl+81)r(C2+82)r(C3+S3)

r(cl)r(c2)r(a3)r(81~82+S3)

Now, cons!dering the transformation u3=u2x~, r:u2 the density of U3=XIX2X3X~ can be round by app1ying

the same technique used above and can be written

down as

(7.18)

(7.19)

a~-l 81T82+83+8~-1 - k h(u3 ) =Cu3 (1-u 3 ) k;O(a)k(1-U 3 ) •

[siEo y(k,j)r(811'82+83+k+j) (l-u )i. . 3

r(8 1+8 2 T8 3+8 4 +k+j)

F( -"3+"~ +S,,;S ,-+S2 +S 3+k +J ;S 1 +S2 +S 3+S" +k+J ;1-u 3 >.1

where

C = r (al -t8 1 ) r (a2 +8 2) r (a 3 +8 3 ) r (a 4 +8 4 )

r{a1)r(a2)r{a3)r{a4)r(81~82+83)

y{k,j) = (-a2 1-a 3+8 3) si (8 1 +8 2+k) si j! (8 1+8 2 +8 3+k)si

34

Substituting the actual values or the parameters, we

obtain

n/2-S SS/2 - k CU 3 (1-u 3 ) k~O(a)k{1-u3)

[_Ëo y(k,j)r(42/10+k+j) (1-u

3)si.

3= r(35/2+k+j)

F(23/10;42/10+k+J ;35/2+k+J ;1-U3 >] where C=r(n/2+1/5)r(n/2+2/5)r(n/2+3/5)r(n/2+4/5)

r(n/2-1/2)r(n/2-1)r(n/2-3/2)r(n/2-2)r(42/10)

(7.20)

y(k,j) = (8/5)j(21/10+k)j

j! (42/10+k). J

(a)k=(6/5)k(7/5)k

k! (42/10)k

Case V. 1/2

For p=6, we find that W 1s distr1buted

as X1X2 X3 where X1 ,X2 and X3 are independent1y

d1str1buted s(n-5, 17/3), s(n-3, 10/3), s(n-1, 1)

respect1ve1y. The density of U2=X1X2X3 can be

written from case IV as

35

n-2 9 CD

feu )=Cu (l-u) kIO (11/3) (10/3) F(5;9+k;10+k;1-u ) 2 2 2 = k k 2

(10)k k!

where C = r (n;-2/3) r (n+1/3) r (n)

9! r(n-5)r(n-3)r(n-1)

Let W1/~U the dens1ty of W 1s given as - 2'

('7.21) h(w) =

c/2 w(n-3J/2(1_w1/2)9kEo (11/3)k(10/3)kF(5;9+k;10+k;1-U2)

(10)k k!

Using (27), we obtain

(7.22) h(w)=

(n-7J/2 1/2 9 CD

c/2 w (l-w) k;O (11/3)k(10/3)kF(1;5+k ;10+k;1-u2)

(10)k k!

(7.23)

36

Case VI. 1/2

For p=8, we find that W is distributed

as XIX2X3X~ where XI ,X2,X3 and X~ are independent1y

distributed Sen-l, 1), s(n-3, 13/4), s(n-5, 11/2),

S(n-7, 31/4) respective1y.

1/2 Let W =R, the density of R can be written down

from case IV as

n-8 33/2 ~ k h(r)=Cr (l-r) k~O(a)k(l-r).

[.Ë y(k,j)r(39/4+k+j) (1-r)jF(23/4;39/4+k+j;35/2+k+j;1-r~

:;-0 r (35/2.,.krj ) J

where

C = r(n)r(n.,.1/4)r(nr1/2)r(n+3/4)

r(n-1)r(n-3)r(n-5)r(n-7)r(39/4)

y(k,j) = (7/2)j(17/4+k)j

j! (39/4rk). :;

(a)k= (3)k(13 /4 )k

k! (39/4)k

We have derived the criterion VI for testing the

hypothesis that the covariance matrices of two

p-variate normal populations are identical; VI

is defined by (3.07) and for p=2, (4.21) becomes

37

(7.24) h

r(nl+hnl-l)r(n2Thn2-l)r(nITn2-l) EVI = r (nl-l)r (n2 -l)r [nl~n2 +h(n 1 +n2 )-1]

wh1ch can be wr1tten as

h (7.25) EVI ::::. r(nl+hnl-l)r(n2+hn2-l)r(nl+n2-2)

r(nl-l)r(n2-l)r[nl~n2-2+h(nl+n2~

r[nITn2-2+h(nl+n2~ r(n l +n2-l)r(1)

r(nl+n2-2)r[nl+n2+h(nl+n2)-~ rel)

wh1ch can be wr1tten as

(7.26) h

EV1 ==

Thus, VI 1s d1str1buted as

where Xl and X2

are 1ndependently d1strlbuted

Let a~b be the two roots of

nI n x (l-x) 2 - V 11-

'"

38

Thus, we wrlte

Uslng (29), (7.27) can be wrltten as

Ia(nl-l, n2-l) +l-Ib(nl-l, n2-l)

-l( (n l +n2-2J/n l +n2 TB nI-l, n2-l) v •

2nl -2 2n2-2 n l +n2 n l +n2

xl (l-XI)

(7.29)

39

In general, the above integral is not easy to evalu

ate but if n 1=n2=m, as for example, then

a:: (1_-Y1_4V1Im ) /2,

b"", ( 1+ -Y1_4v1Im ) /2 = 1-a

so that

Then (7.28) becomes

-1 1 P(Vl~v} = 2 la(m-l, m-l} ~ 2 B (m-l, m-l) v1 - 1 m.

log 1~-y1_4vllm

l_-Y1_4v1Im

We notice that E. S. Pearson and S. S. Wilks ~~

have given this in another forme We will now

consider the evaluation of the density, and the

author has not seen the application of Consul's

formulae for this criterion in the literature;

hence, the author assumes that these results are

new.

Case l. For p=2, the hth moment of Vl has been

evaluated as

(7.30) h

EV I = r (nI Thnl -1)r (n2 Thn2-1 )r (nl +n2-l )

r(nl-1)r(n2-1)r[nl+n2+h(nl+n2)-~

Let nI == n2 = d then~ (7.30) becomes

h (7.31) EVI ==r(2d-1) r(dh+d-1) r(dh+d-l)

(2dh+2d-l) r 2 (d_1)

Using (19)~ (7.31) becomes

h (7 .32 ) EV I = r (2d -1 )

~--~------~~--~~

40

2 -1/2 2d-1-1/2 r (d-1)(2w) 2

• r(dh+d-l)r(dh+d-1) 2dh

2 r(dh+d-1/2)r(dh+d)

Using the resu1t (30) obtained by Consul ~~, we

obtain

(7 .33) f (V t> .:=

d In (30), let x=4 V , s=h~ p=d~ a~d-l~ b=d-l~ m:1,

I

n:l, and we obtain 2

(7.34) f(V I).: r(2d-l)/ir

r2 (d_1)22d- 2

Vl-1(4dVI) (d-1J/d ~_(4dVI)1/dJ1/2F(1;1/2;3/2;1_(4dVI)1/d) d r(3/2)

(7.36)

(7.37)

(7.38)

41

which simplifies to

2r(2d-l) • Vl-lld(1-4Vllld)112F(1;1/2;3/2;1-4Vllld)

dr 2 (d_1)

Case II. For p=4, (4.21) becomes

h EVl

_

2 .Hl r (ni"1-2j )

3=

2 • j~l r(n l i"hn l +1-2j)r(n2 +hn2 +1-2j)

r (n-hntl-2j )

Let n =n %d, then (7.36) becomes 1 2

h EV -1 -

2 j~l r(2d~1-2j) • r(dh+d-1)r(dh~d-3)r(dh+d-1)r(dhTd-3)

2 r (d~1-2j) r(2dh+2d-l)r(2dh+2d-3)

Using (19), (7.37) becomes

h EV l =

r(dh+d-3)r (dh+d-1)r (dh+d-3) 2-4dh

r(dhi"d-1/2)r(dh+d)r(dh+d-3/2)

(7.39)

Using (31) with x = 16dv l' s=h, q=d, a=d-3, b=d-3,

c=d-1, r=d, m=5/2, n=3/2, p=1, we obtain

f(V 1) =

K V -4/d(1_16V l/d)~. t (3/2).(5/2). (l-16V l/d)i 1 1 i=O 3 3 1

j! r(5 ... j)

[ l/d -l/d ] .F 1;3;5+j;-(l-16V

1 )V

1 116

where K:::.n r(2d-1)r(2d-3) 2 2

1024d r (d-1)r (d-3)

42

CHAPTER 8

THE EXACT DISTRIBUTION FOR

THE SPHERICITY TEST IN

THE MOST GENERAL CASE

In (5.04), we have evaluated the density of W=À2/ N

where À is the likelihood ratio criterion for

testing sphericity, we have obtained

(8.01) f(w) =

43

p-l,O [ 1 n/2-1+1/p,n/2-1+2/p, ••• ,n/2-1+(P-l)/P] K Gp-l p-l w

P , n/2-1-1/2,n/2-1-2/2, ••• ,n/2-1-(p-l)/2

where Kp= r(np/2) (2w)(p-1)/2

p(pn-l)/2.~ r[(n+l-i)/2] 1.=1

Some expansions of G-function are available in the

literature but, due to the special nature of the

parameters n/2-1-l/2, ••• ,n/2-1-(p-l)/2, it appears

that none of the expansions available in the litera

ture can be used to expand (8.01) in a series so

that the percentage points can be computed.

We will use an article by Mathai and Rathie ~~ in

which they give the density of W in simple algebraic

functions by using the residue theorem.

44

In order to determine the density, we will evaluate

the residues at the poles of the Gamma products in

(5.02). The Gamma products in (5.02) excluding

K is, P

(8.02) r(œ-i)r(œ-l)r(œ-3/2 ••• r[œ-i(p-2)] r[œ-l(p-l~

r(œ+l/p)r(œ~2/p) ••• r[œ+(p-2)/p]r[œ+(p-l)/P]

(8.03)

where œ=s+in-l. The poles of the alternate Gammas

in the numerator of (8.02) coincide, whereas the

poles of the adjacent Gammas do not coincide. So

we will separate the two types of poles and repre

sent aIl the poles in two sets. For simplicity, we

will consider the cases p-odd and p-even separately

and further when p=2 the problem is simple and hence

we consider only the cases where p>2.

Case 1. p-odd (p>2)

The poles of (8.02) are obtained from the various

factors in (8.03) and (8.04) by equating them to

zero and the indices represent the orders of the

poles.

. . .

(8.05)

45

Now for example, cons1der the evaluat10n of the

res1due correspond1ng to a pole of order j obta1ned

from a factor of the type (a-ip+1)i, j=I,2, ••• ,

i(p-l)-l, j=1. The res1due at s~l-in+ip-1 1s

obta1ned as,

a· . (w) = 1,3

l i-1

tS [ i 8-1 -8] (a-ip+1) E(W ) w

at s=l-in+ip-1.

Hence from (8.03) we will get the res1dues,

a .. (w), (1, j ) E a and a .. (w), ( 1, j ) E a' 1,3 1,3

where

a = {(1,j) j=1, 1=1,2, .•• ,i{p-I)-I} and

a'= {{1,j) j=i{p-l), 1=i{p-I), i{p+l), .•. ... } .

S1m1larly, we will denote the res1dues com1ng from

(8.04) by b .. (w). That 1s, 1,;)

(8.06) b •• (w) = 1,J

l i-1 o ';-1 (j-1)! osU .

at S = 1-!n'f"! (p+l )-i

where,

46

(i,j)E:bvb', b:{(i,j) 1 j.::i, i.::l,2, ••• ,i(p-l)-1} and

b'={(i,j) 1 j::::i(p-l), i==i(p-l), i(p+l), ••• . .. } .

In the following discussion we will use the general

notations A. and B. where 1, 1.

i s-l -8 A. =(Cl-p/2+i) E(W ) w , 1,

Bi ~ ô log [(a-PI2+i);Ï E(W8-1

) w -8] oS

while evaluating a .. (w) and b .. (w) will be evaluated l.J l.J

with the help of C. and D. given below. 1. 1,

Ci = (Cl-i{p+l }+i)i E (Ws - 1 ) w- s ,

D i~ ~ log [ (a-Hp+l}+i)j E(W8-

1 ) w-8.]

oS

Thus we have

·e

47

(8.07) ô ( 1 ) A.::A. _ A.B.

1. 1. - 1.1.

ôs

and

A .(m) A.(m-l)B. (m-l) (m-2) (1) m-l A.B.(m-l) (8.08) 1. = 1. 1.~ 1 Ai Bi ~···+(m-l) 1. 1.

where A.(P) and B.(P) denote the r-th derivatlve or 1. 1.

A. and B. wlth respect to s respectively. (8.08) 1. 1.

gives a recurrence relation and further a .. (w) and 1.3

b .. (w) can be calculated easily with the help of 1,3

A B B (p) C D and D .(p). where • , • , • , • , • , 0'" 01, 01, 01, 01. 01. ~

A ._A. at a=~p-i, Co':=C,: at a=i(p+l)-i, 01, - 1, ., .,

B._B. at a=~p-i, D ._D. at a=~(p+l)-i, 01,- 1, 01,- 1,

B .(p) B.(P) at a;::-ip-i and D (p) D.(P) at a= i(p+l)-l. 01, == 1, oi = 1,

Hence in the following discussion we will evaluate

these quantities in the various cases. Now consider,

i = 1,2 , ••. , i (p-l ) -1 •

48

The other Gammas in (8.02) are unaffected when

mu1tip1ied by (œ-ip+i)i. Now distribute one factor

(œ-!p~i) each to the Gammas r(œ-!p+i), r(œ-!p+i-1),

..• ,r(œ-ip+1), thus absorbing aIl the factors.

Thus (8.09) can be written as,

(8.10) ~(œ-ip+i+1~ i+1 r (œ-ip+i+2) ••• r(œ-!p+!{p-1})

(8.11)

(8.12)

1 , 2 , i-1 (œ-~p+1)(œ-~p+2) ••• (œ-~p+i-1)

Therefore,

A. [r(œ-!p+i+1Y i+1 r (œ-!p+i+2) ••• r(œ-!p+!{P-1}) • 01.= ~

~--------~-------------------------------1 , 2 , i-1 p-1

(œ-~p+1)(œ-~p+2) ••• (œ-~p+i-1) .nr(œ+j/p) ;):=1

• [r(œ-1) .•• r(œ-!{p-1})] w- s at œ::!p-i

(_1)i(i-1J/2 (P:~J/2-ir(j)l(Pn1J r(-i-!+j) ln-1-lp+i w

_ ;)=1 j=l - --------------------------~------------------------p-1 [ ] .n r(!p-iTj/P) (i-1)!(i-2)! ••• 1! ;)=1

Now,

B. ô 1.=

ôs

(8.14)

(8.15)

j(P-1J[ . ij p-1 .t -y+t «(I-i {p+l }+j) - .t [-y+t (a+j Ip)1 -J=l J=l 1

i-1 .t j/(a-ip+j) -log w, J-1

· 49

where y is Euler's constant (32) and t(.) is defined

in (33).

p-1 i-1 _ .t t~a+j/p) - .t j/(a-ip+j) -log w.

J=l J=l

j(p-1J p-1 + t g(r+l,a-i{p+l}+j)- t g(r+l,a+j/p)

j=l j=l

+ t j/(a-ip+j)P+ i-1 1 ]

j=l

where g(.) is defined in (34).

While evaluating Aoi' Boi and Boi(~J for particular

values of p the Gammas with the negative arguments

can be converted by using the formula

r(z) r(l-z)= w/sin(wz) and (l+a) = (-l)ml(-a)m. -m

C D and D .(~J can be calculated in a similar oi' oi 01-

fashion,

(8.16)

50

Now, by using the above notations the density function

f(w) of W can be written as,

f(w)= K [~ , a .. (w) + ~bt.Jb' b.;.;(w)l, O<w<l. p l. ava 1.;) l. "u IJ

Case II. p-even (p>2)

8-1 In this case E(W ) exc1uding Kp and after cance1ing

aIl the common factors, becomes,

(8.17) r(a-1)r(a-3/2) .•• r(a-~{p-2})r(a-~{p-1})

(8.18)

(8.19)

r(a+1/p) ••• r(a+{~p-1}/p)(a-~)r(a+{~p+1}/p) ••• r(a-i{p-1})

;(p-2) ;(p-2) /1 1 ==.n r (a-~p+j ), /1 2 ==.n r (a-~{p+1 }+j ) and

;)=1 ;)=1

p-1 /1 3 = (a-~) j~1 r(a+j/p).

j:f:ip

According to our notation,

A A (p) Proceeding as before, we get . , ., B ., B. , 1. 01. 01. 01-

C ., D ., and D .(p). For examp1e, whi1e ca1cu1ating 01. 01. 01.

a .. (w), (i,j )ea, we have, 1.;)

51

(8.20) Ai=~(a-lp~i+1~ i+1 r (a-lp+i+2) ••• r(a-lp+l{p-2})â2 W- S

(8.21)

2 i-1 (a-lp+1) (a-lp+2) ••• (a-lp+i-1) â

3

Now, Aoi is obtained by evaluating Ai at a=lp-i.

Incident1y, the structure of the density function

when p is even, remains more or less the same as

in (8.16) and it is given below.

f(w)=K [1: ,a .. (w).,. 1:b

b' b" b •• (W)], O<w<l, p aua 1,;] v v 1,;]

where the terms a· .(w), b .. (w) as weIl as the sets 1,;] 1,;]

a, a', b, b', b" are given in the fol1owing tables.

e

aij(w) =

A . 01-

Boi

B ('1') oi

e

T A BLE 1

a (w) in the dens1ty funct10n for the case p-odd (p>2) ij

e

l Ai(j-l), A.z:(l)= A.z:B.z:' (1,j)ta, a_{(1,j) 1 j=1,1.l111, ••• ,~(p-1)-1}

(j -1) 1

j(p-l)-.z: j(p-l) (_1)i(i-l)/2 n r(j) n r(-i-1+j)

j=l j=l win- i P-l+i

p-l n r(!p-1 ... j/p) (1-1)1 (1-2)1 ••• 1!

j=l

;(p-l) i(p-l) p-l i-l t t(-1 ... j)+ t t(-i-1+j)- t t(ip-1+j/p)- t j/(j-1) -log w.

j=i+2 j:::l j=,l j=l

[ i(p-l) i(p-l)

(_l)'1'-lrl (1+1)g(r+l,1)+ t g(r+1,j-1)+ t g(r+l,j-1-!) j=.z:+2 j=l

p-l i-l] - t g(r+1,!p-1+j/p)+ t j/(-1_j)'1'+1

j=l j~l

\JI 1\)

e e

T A BLE 2

a .. (w) in the density function for the case p-odd (p>2) ~J

a .. (w), (i,j)€a', a': {(i,j) 1 j = !(p-l), i;;::. !(p-l), !(p+l), ••• } ~J

A . O~

BOi

B (p) oi

(_1)i(p-1)(i-i{p+l}) j(~-1) r(-!-i+j) j=1

------------------------------------------------p-l

n r(!p-i+j/p) (i-l)l (i-2)1 ••• (1-!{p-l})! j=l

", jn - ip-l +{,

j(p-1) p-l j(p-3) L t(-!-i+j)- L t(!p-i+j/p)- L j/(-i+j)

j=1 j=1 j=l

i-1 -!(p-l) L l/(-i+j) -log w

j=I(p-l)

1 [ l(p-1) p-1 (-l)P- r! !(p-l)g(r+l,l)+ L g(r+l,-!-i+j)- L g(r~l,!p-i+j/p)

j=l j~l

+ L j/(-i+j)P+ +!(p-l) L l/(-i+j)P+ ;(p-3) 1 i-1 1 1 j=1 j=j(p-1)

•

\Jl lA)

e e

T A BLE 3

b .. (w) in the density function for the case p-odd (p>2) 1,J

e

(j-1J (1) b .. (w) _ 1 C. , C. _ C.D., (i,j)eb, b={(i,j) 1 j=i, i=1,2, ••• ,l(p-l)-1} 1,;) - 1, 1, - 1, 1,

coi

D . 01,

D (fi) oi

(j-l>t

(_1)i(i-1J/2 ;(Pn1J - i r(j) ~1

;(R-1J r(l_i+j) j=1

p-1 fi r(l{p+l}-i+j/p) (i-l)! (i-2)! ••• l!

j=1

in-;p-3/2+-i w

i(pt 1J t(-1+j)~ j(Pt1J t(l-i+j)- pt 1 t(l{p+l}-i+j/p) j:i~2 j=1 j~1

i-1 - ~ j/(-i+j) -log w

j=1

[ ;(p-1J ;(p-1J

(_1)fI-1 r ! (i~l)g(r+l,l)~ t g(r+l,j-i)~ t g(r~l,l-i+j) j==i+2 j=1

p-1 i-1] - ~ g(r+l,l{p+l}-1+j/p)+ ~ j/(_1+j)fI+-1 ~1 ~1

'"" .1::"

e e e.

T A BLE 4

bij(W) in the density function for the case p-odd (p>2)

bij(W), (i,j)e:b', b' == {(i,j) 1 j=~(p-l), :1;::~(p-l), !(p.,.l), ••• }

Coi

Doi

D (ft) oi

(_l);(p-l)(i-i{p+l}) j(~-l) r(!-i+j) j=l

p-l rr r(!{p+l}-i+j/p) (i-l)l (i-2)l ••• (i-!{p-l})l

j~l

win- i p-3/2i-i

j(p-l) p-l j(p-3) E t(~-iTj)- E t(~{p.,.l}-itj/p)- t j/(-i+j)

j=l j~l j-l

i-l -~(p-l) E l/(-i+j)- log w

j~i(p-l)

1 [ i (p-l) p-l (_l)ft- rt !(p-l)g(r.,.l,l)+ E g(rtl,!-i+j)- E g(r.,.l,l{p+l}-itj/p)

j-l j~l

+ t j/(_itj)ft+ + !(p-l) E l/(_i+j)fti-;(p-3) l i-l l ']

j=l j=;(p-l)

\J1 \J1

o

e

aij

(w) _

Aoi

Boi

B (roJ oi

e

T A BLE 5

a .. (w) ln the denslty functlon for p-even (p>2) ~J

e

1 A.fj-1J, A. (1~ A.B., (l,j)€a, a= {(l,j) 1 j=l, 1=1,2, ••• ,lp-2} (j-1) t ~ ~ - ~ ~

J'(' 1) ;(p-2) ;(p-2) (-1);§~ ~- n r(j-l) n r(-1-1+j)

j=i~2 j=1 in-ip-1.,.i w .

p-1 (p/2-1-i) fi r(p/2-1+j/p) (1-1)t(1-2)t ••• 1! j=1 j:f;P

l(p-2) j(p-2) p-1 t t(-l+j)+ t t(-l-l+j)- l/(ip-l-i)- E t(ip-l+j/p)

j=i~2 j=1 j=1 3*ip

i-1 - E j/(j-l) -log w

3=1

(_1)~-1r! (1+1)g(r+1,1)+ E g(r+1,-1+j)+ E g(r+1,-1-1+j) [ ; (p-2J i (p-2J

3=.;,.,.2 3=1

1 p-l ';'-1 1J - l/(ip-l-i)ro+ - E g(r+1,ip-l+j/p)+ E j/(j_l)ro+ j~1 j=1 3:f.ip \J1

0\

e e

T A BLE 6

a .. (w) ln the denslty functlon for p-even (p>2) ~3

e

aié/(w), (l,j)e:a', a':::= {(l,j) 1 j=lp-1, 1=lp-1, lp, ••• }

Aoi

(_1);(p-2)(i-lp) i(R- 2) r(-1-1~j) win-;V- 1+i é/=l

p-1 (lp-l-1) fi r(lp-l~j/p) (1-1)t(1-2)1 ••• (1-l{p-2})1

é/=1 é/=I=;P

;(~-2) p-1 i-1 ~ t(-l-l+j)- t t(lp-l~j/p)- 1/(lp-l-1)- 1(p-2) t l/(j-l)

é/=1 é/=1 é/=;(p-2) é/#;p

BOi j(~-4) j/(j-l)- log w é/=1

B (p) oi

[ ;(p-2)

(-1)p-1rI 1(p-2)g(rr1,1)+ t é/=1

i-1 p+1 1( -2) t + 1/(lp-l-1) ~ 2 p é/~;(p-2)

p-1 g(r~l,-l-l+j)- t g(r~l,lp-l+j/p) é/:::.1 ..

é/:I=;p

1/(j_l)P+1+ t j/(j_l)p+1 ;(p-4) 1 é/=1

\.on -'1

e

bid(W) ==

o . o~

D • o~

D (ft) oi

e e

T A BLE 7

btd(W) in the density funotion for p-even (p>2)

1 0.'d- 1 ), 0.(1)_ O.D., (i,j)&b, b= {(i,j) 1 j=i, i c l.2, •• .,ip-l} ~ ~ - ~ ~

(j -1) 1

(_1);i(i-1) j(~-2)r(i_1+j) j(~-2) r(-1+j) win-;p-3/2+i d~1 d=i+2

p-1 (!(p+l}-1-!) n r(i{p+l}-1+j/p) (1-1)1(1-2)1 ••• lt

3=1 3:t:;P

l(p-2) ;(p-2) p-1 t ~(-1+j)+ t ~(!-1+j)- t ~(!{p+1}-1+j/p)- 1/(i{p+l}-1-!)

3:i+2 3=1 3~1 3~ip

';'-1 - t j/(j-1)- log w

3=1

ft-1 [ ; (p-2) ; (p-2) (-1) r1 (1+1)g(r+1,1)+ ~ g(r+l,j-1)+ t g(r+l,!-1+j)

3=~+2 3=1 p-1

- t 3=1 j=/:ip

';'-1 ft+1] g(r+l,!{p+l}-1+j/p)+ 1/(!{p+l}-1-!)+ t j/(j-1) j::: 1

\11 00

e e

T A BLE 8

b (w) in the density function for p-even (p>2) ij

b .. (w), (i,j)eb', b'={(i,j) 1 i=j=!p} .""

c . 0."

D . 0."

D (1') oi

;(p-2) • . (_1)P(p-2)/8 fi r(i-!p+j) win-;p-3/2+'"

j=l

p-l fi r(!+j/p) (!p-1)1(!p-2)1 ••• Il

j:::l j=l:;p

;(p-2) p-l ;(p-2) t t(!-!p+j)- E t(!+j/p). l j/(!p-j)- log w

j=l j=l j=l j~;p

1 [ ;(p-2) p-l (_1)1'- r! (!p-1)g(r+1,1). l g(r+1,i-ip.j)- E g(r+1,i.j/p)

j=l j~l

+ ;(~-2) j/(_!p+j)1'+l] j:: 1

j:t:;p

e

U'I \0

e e e

T A BLE 9

bij(W) in the density function for p-even (p>2)

b .. (w), (i,j)eb", bIt == {(i,j) 1 j=~p-l, i=~p+l, ~p+2, ••• } 1-J

c . 01-

D . 01-

D (r) oi

(_1)j(p-2)(i-lp) 1(~-2) r(!-i+j) wln-Ip-3/2+i j=l

p-1 (~{p+l}-i-~) fi r(~{p+l}-i+j/p) (i-l)!(i-2)! ••• (i-!{p-2})!

j=l j=l=-;P

i(~-2) p-1 ~ ~(~-i+j)- l/(~p-i)- E ~1 ~1

l(p-4) ~(~{p+l}-i+j/p)- E j/(j-i)

j=l j~lp

i-1 -!(p-2) E l/(j-i)- log W

j~I(p-2)

(_l)r- r! ~(p-2)g(r+l,1)+ E 1 [ j(p-2)

j::::1

p-1 - E

if:;; 1 j~jp

;(p-4) g(rTl,!{p+l}-i+j/p)+ L

if=1

g(r+l,~-i+j)+ 1/(~p_i)r+1

1 i-l ] j/(j_i)r+ T !(p-2) E, 1/(j_i)r+1 if=j(p-2)

0\ o

e

p=3

p=4

p=5

e

T A BLE l a

PARTICULAR CASES

f(w) ~ K [t A • ~ t COi] 3 i=l 01- i=l

where the terms are given in Tables l, 2, 3, and 4, and K3 is glven

in (8.01).

f(w) = [ ~ ~

KEA • ~ C + C D + E 4 i=l 01- 01 02 02 i=3 C .l 01-

e

where the terms are glven ln Tables 5, 6, 7, 8, and 9, and K4 ls glven

ln (8.01).

f(w) .= [ ~ ~

K A + E A . B . r C + E 5 01 i~2 01- 01- 01 i~2 C • D .] 01- 01-

where the terms are glven ln Tables l, 2, 3, and 4, and K ls glven 5

ln (8.01).

0\ .....

e

p=6

p=.7

p=8

e e

f(w)= K [A 1+ ! A .B .+ C 1+ C nD n+ C ~ (D2~~ D(~) )~ ! C .D .] 6 0 i~2 O~ O~ 0 00 00 OQ OQ OQ i~4 O~ O~

where the terms are given in Tables 5, 6, 7, 8, and 9, and K6 ls glven

in (8.01).

f(w)==K [A ~ A B of- i' A .( B2 .+ B(~») + C + C D 7 01 02 02 i=3 O~ O~ O~ 01 02 02

+ Ë C • ( D2

• of- D ( ~ ) ) 1 i=3 O~ O~ O~

where the terms are given in Tables l, 2, 3, and 4, and K7 is given

in (8.01).

[

w 2 (1) f(w)~ Ka A 1r A nB n+ E A ~ (B ~+ B ~ )+ C 1+ C 2D n o 00 00 i~3 OQ OQ OQ 0 0 00

2 (1) 3 (1) (2) + C03 (D03+ D03 )+ C04 (D04+ 3 D04 D04+ D04 )

-t E C • (D .+ D. ) w 2 (1) ]

i~6 O~ O~ o~

where the terms are given in Tables 5, 6, 7, 8, and 9, and K is given a in (8.01). 0\

N

CHAPTER 9

APPLICATIONS

In this chapter, we illustrate, by means of examples, the

problems which can be solved by applying the theory devel

oped in the thesis.

For testing sphericity, our criterion will be M = Isl sP

where S is the sp matrix and s = 1 tr S. For large N, p

the hypothesis is to be rejected at approximate level of

significance a if

where

f = lp(p+-l)-l, À

2

2 _ (2p -t-p+2)/6p.

On the basis of samples of size N from N (pi ,Li), i=1,2, i

to test the hypothesis that the dispersion matrices are

identical, the statistic to be used is

A A

where v. = N.-l, v = v +v , t S~/v~, t = (SlrS2)/V. 1. 1. 1 2 i- ....

When samples are large in size, the hypothesis is to be

rejected at approximate level o~ signi~icance a i~

T = -plnB > x~(a)

where ~ = Ip(p+l) 2

p == 1-2

2p +3p-l •

6(p+l)

64

Example 1. Two measurements were made on 15 students:

psychological test, achievement test.

Xl psychological test X2

achlevement test

68.00000 61.00000

67.00000 62.00000

42.00000 48.00000

58.00000 50.00000

72.00000 72.00000

75.00000 76.00000

70.00000 42.00000

47.00000 70.00000

52.00000 55.00000

93.00000 123.00000

71.00000 65.00000

69.00000 73.00000

86.00000 110.00000

63.00000 77.00000

79.00000 86.00000

65

The above data is an exercise given in the "Handbook of

Methods of Applied Statistics", Vol. l, by Chakravarti, Laha,

and Roy, Wiley, New York, 1967.

For this example, we will calculate the follow1ng:

1)

x=

X Il

X 12

. . . x· IN

x x ••• XpN pl p2

, a p.N matr1x

2) XX'=p.p matrix, X' = transpose of X.

3) N 1: xl. i==1 1..

•

• N 1: X •

i-1 p1.

4) 1 YY', a p.p matr1x N

, a p.l matr1x

5) S ~XX'-l YY', a p.p matrix N

6) M = Isl ---(trS)P

66

x _ (68 67 42 58 • • • 79) 61 62 48 50 86

X'=

XX' =

68

67

42

58

•

79

(75145 00 0009 80458.00006

61

62

48

50

86

y = (107700)

1148.0

172495056259 1 YY' = N

77274.75006

(2649043750 s

3183.25000

M = 0.3549178635

80458000006)

89210.00006

77274.75006

82369.00009

3183025000)

6841.00000

67

1

68

f=2, À=l, 2

T~14.28> x2

(5%)= 5.99

Hence, we reject the hypothesis.

Examp1e 2. We give the sepa1 1ength, sepa1 width, petaI

1ength and petaI width in mi11imeters of 15 f10wers of each

of these two specie~: Iris Setosa and Iris Versico1or. In

this examp1e, we will test for sphericity in both species.

For Iris Setosa, we have

58.00000 40.00000 12.00000 2.00000

36433.00003

24677.00001

10483.00000

1537.00000

36211.26672

1 YY'= 24517.53335 N

10465.40001

1523.13333

24677.00001

16765.00001

7102.00000

1050.00000

24517.53335

16600.06668

7085.80000

1031.26666

10483.00000

7102.00000

3055.00000

449.00000

10465.40001

7085.80000

3024.60000

440.20000

1537.00000

1050.00000

449.00000

71.00000

1523.13333

1031.26666

440.20000

64.06666

• e e

l RIS SET 0 S A l RIS VER SIC 0 LOR

SEPAL PETAL SEPAL PETAL

Length Width Length Width Length Width Length Width

51.00000 32.00000 14.00000 2.00000 70.00000 32.00000 47.00000 14.00000

49.00000 30.00000 14.00000 2.00000 64.00000 32.00000 45.00000 15.00000

47.00000 32.00000 13.00000 2.00000 69.00000 31.00000 49.00000 15.00000

46.00000 31.00000 15.00000 2.00000 55.00000 23.00000 40.00000 13.00000

50.00000 36.00000 14.00000 2.00000 65.00000 28.00000\ 46.00000 15.00000 '~

"

54.00000 39.00000 17.00000 4.00000 57.00000 28.00000 . 45.00000 13.00000

46.00000 34.00000 14.00000 3.00000 63.00000 33.00000 47.00000 16.00000'

50.00000 34.00000 15.00000 2.00000 49.00000 24.00000 33.00000 10.00000

44.00000 29.00000 14.00000 2.00000 66.00000 29.00000 46.00000 13.00000

49.00000 31.00000 15.00000 2.00000 52.00000 27.00000 39.00000 14.00000

54.00000 37.00000 15.00000 2.00000 50.00000 20.00000 35.00000 10.00000

48.00000 34.00000 16.00000 2.00000 59.00000 30.00000 42.00000 15.00000

48.00000 30.00000 14.00000 1.00000 60.00000 22.00000 40.00000 10.00000

43.00000 30.00000 Il.00000 1.00000 61.00000 29.00000 47.00000 14.00000

58.00000 40.00000 12.00000 2.00000 56.00000 29.00000 36.00000 13.00000

Source: R. A. Fisher. The use of multiple measurements in taxonomie problems. Anna1s

of Eugenies (London), Vol. 7, p. 179.

0-\C

221.73330 159.46665

159.46665 164.93333 8

1 c::.

17.59999 16.19999

13.86666 18.73333

1811 = 895254.1464843750

M = 0.0070912531

f=9, À=1.5, 2

T=66.96> x (5%)= 16.9 9


For Iris Versico1or, we have

X'=

70.00000

64.00000

69.00000

55.00000

56.00000

32.00000

32.00000

31.00000

23.00000

29.00000

17.59999

16.19999

30.39999

8.79999

47.00000

45.00000

49.00000

40.00000

36.00000

13.86666

18.73333

8.79999

6.93333

14.00000

15.00000

15.00000

13.00000

•

•

13.00000

XX' =

54123.95323

25163.97270

38458.96104

12056.98830

25163.97270

11806.98830

17907.98052

38458.96104

17907.98052

27404.97661

5651.99415 8594.99221

70

12056.98830

5651.99415

8594.99221

2719.99805

1 YY'= -N

53520.96104

24908.75395

38050.06259

11946.04651

602.99231

255.21878

408.89849

110.34181

24908.75395

11592.57619

17708.56645

5559.99024

255.21878

214.41214

199.41409

92.00392

IS21 = 46502624.1093750001

M = 0.0052951742

f=9, À=1.5, 2

T= 71.55> xg (5%) = 16.9


71

38050.06259 11946.64651

17708.56645 5559.99024

27051.21488 8493.31838

8493.31838 2666.6626

408.89849

199.41409

353.76178

101.67384

110.34181

92.00392

101.67384

53.33545

Examp1e 3. Using the same data as in Examp1e 2, we will test

the equa1ity of covariance matrices. We have

... 1:1== SI -

14

43.07087

18.22991

29.20703

7.88155

15.83809

11.39047

1.25714

0.99047

18.22991

15.31515

14.24386

6.57170

11.39047

11.78095

1.15714

1.33809

29.20703

14.24386

25.26869

7.26241

1.25714

1.15714

2.17142

0.62857

7.88155

6.57170

7.26241

3.80967

0.99047

1.33809

0.62857

0.49523

29.45448 ...

14.81019 t = Sl~ S2 = 28 15.23208

4.43601

1 t l 1 = 1210.5139017105

It 1 = 23.3032912015 2

Iii = 563.6491687297

14.81019

13.54805

7.70050

3.95490

B = 0.000000000000001893125

f=10, p= 711 , 340

2 T=28.64> x10

(5%) = 18.3

15.23208

7.70050

13.72006

3.94549

72

4.43601

3.95490

3.94549

2.15245

Hence, we reject the hypothesis at the 5% 1evel but we cannot

reject at the 1% level.

73

Examp1e 4. He1ght and we1ght of 16 tw1ns of oppos1te sex.

MALE FEMALE

He1ght We1ght He1ght We1ght (cm) (kg) (cm) (kg)

136.00000 27.20000 132.00000 29.40JOO

141.00000 35.00000 133.00000 29.00000

137.00000 30.60000 140.00000 27.80000

137.00000 30.40000 135.00000 26.80000

134.00000 33.20000 131.00000 31.20000

134.00000 31.20000 137.00000 30.20000

130.00000 28.40000 133.00000 27.80000

139.00000 30.00000 140.00000 28.80000

128.00000 27.00000 123.00000 26.40000

132.00000 27.80000 136.00000 30.00000

133.00000 28.40000 134.00000 28.00000

129.00000 26.60000 134.00000 28.40000

137.00000 30.60000 134.00000 29.90000

128.00000 24.40000 134.00000 28.60000

134.00000 30.00000 135.00000 26.40000

130.00000 27.40000 127.00000 27.80000

The above data 1s an exerc1se g1ven 1n the "Handbook of

Methods of App11ed Stat1st1cs", Vol. l, by Chakravart1, Laha,

and Roy, W11ey, New York, 1967.

In this examp1e, we will test for sphericity in the male

and fema1e populations. For the male population, we have

X'=

136.00000

141.00000

137.00000

137.00000

•

130.00000

__ (286195.00024 XX'

62662.39990

y _ (2139.0) 467.7

27.20000

35.00000

30.60000

30.40000

•

27.40000

62662.39990)

13783.47997

_(285957.56286

1 yy' N

62539.01242

62539.01242

13677.30245

( 237 Il ~'7r=n

...... J,.J v

123.38748

123•38748)

106.17752

Is1 1 = 9986.0544662475

l

74

M = 0.3383059543

f=2, À=l,

T = 16.20> x2 (5%) = 5.99 2


For the fema1e population, we have

X'=

132.00000

133.00000

140.00000

135.00000

•

127.00000

__ (285960.00024 XX'

61020.19992

y=(2138.0) \ 456.4

29.40000

29.00000

27.80000

26.80000

•

27.80000

61020.1999 2 )

13053.48998

285690.25024 60999.81242 1 YY'= N

60999.81242 13024.51557

75

8 = (269.75000

2 20.38751

20.38751)

28.97440

1821 = 7400.1946659088

M = 0.3317124187

f=2, À=l,

T= 16.50> X~(5%) = 5.99


76

Example 5. In this example, we will test the equality of

covariance matrices of the male and female populations. We

have

t1=Sl =(15.82916

15 8.22583

t 2= S2 = (17.98333

15 1.35916

t= Sl~82 = (16.09625

30 4.79249

1 il 1 = 44.382465

It21 = 32.889750

1;: 1 = 53.195685

8.22583)

7.07850

1.35916)

1.93162

4. 79 249) 4.50506

B = 0.0000487290

f=3, p=167 , 180

2 T= 0.66< x (5%) = 7.81 3

77

Hence, we accept the hypothesis at the 5% 1eve1 and a1so at the

10% 1eve1.

CHAPTER 10

CONCLUSION

Although this thesis is in part expository, it contains

some new results and also modi~ications o~ proofs of other

known results.

78

The first hypothesis considered - sphericity - seems richer

in the sense that it has yielded more results than the

second - equality of covariance matrices - •

One of the reasons for this fact is that the literature

contains more relevant papers on the ~irst hypothesis than

on the second.

The author was, thus, able to read Consu1's article on

sphericity and from there develop a simp1er method for

getting exact distributions. The author recognizes the

fact that he did not pursue beyond the value p=8, although

it wou1d have been possible, since the densities obtained

were complicated enough. The theory developed by Mathai and

Rathie in chapter 8 compensates for the forementioned.

The author was able to apply Consul's formu1ae to a few

particular cases of the criterion used when testing equality

o~ covariance matrices. However, it wou1d be useful to

develop ~ormulae o~ this type when the numerator and

denominator do not contain an equal number of Gammas.

AIso, it would be worthwhile to try applying Mathai and

Rathie's method to this criterion.

79

80

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.i

the distribution of the likelihood ratio criterion for...

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