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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
Hence, we reject the hypothesis.
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
Hence, we reject the hypothesis.
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
Hence, we reject the hypothesis.
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
Hence, we reject the hypothesis.
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
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