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RLAD INSTRUC'TIONSREPORT DOCUMENTATION PAGE • OkH (OMPI EriO IoHMII. 8N -T ACCESSION No. 3. REC a66 w-8 7 1 9 5.19 r~u~

4. TITLE (endi Suballej S. TYPE OF REPORT & PEIkIOU COVERED

Admissible Bayes Tests for Structural T.el" - Rpt - Sept. 1987Relationship 6. PERFORMING ONG. REPORT NUMULN

87-31C -U"TN ii T (0 1. CONTRACT OR GRANT NUMBER(.)

Wei-Hsiung Shen and Bimal K. Sinha F49620-85-C-0008."~.

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Structural relationship, admissible Bayes test, likelihood ratio test,prior distribution.

20 AD% 1 RAC T (Ci.tllnuo on reverse 4dao II noc.e.o.q and Idenaiy by block numb*,)

It is an open problem to construct a test for structural relationshipamong the mean vectors of several multivariate normal populations withknown but unequal covariance matrices. In this paper, a class of admissibleBayes tests for the above problem is derived. As a byproduct, in the specialcase of known and equal covariance matrices, the likelihood ratio test ofRao(1973) is shown to be admissible Bayes.

DD IIA.) 1473 UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE 1When DOIo Enre'odI

A.Th. ?87-1959

ADMISSIBLE BAYES TESTS FOR STRUCTURAL

RELATIONSHI P

Wei-Hsiung Shen and Bimal K. Sinha*

Center for Multivariate AnalysisUniversity of Pittsburgh.

Piltchtirnh Pa Ir912n

Center for Multivariate Analysis

University of Pittsburgh

W-7 12 29 21?

ADMISSIBLE BAYES TESTS FOR STRUCTURAL

RELATIONSHI P

Wei-Hsiung Shen and Bimal K. Sinha*

Center for Multivariate AnalysisUniversity of Pittsburgh

Pittsburgh PA 15260

September 1987

Technical Report Number 87-31

A e C 3r, ,V

L'

Center for Multivariate AnalysisUniversity of Pittsburgh

Pittsburgh PA 15260

S-

*Research supported by the Air Force Office of Scientific Research (AFOSR),under Contract F49620-85-C-0008. The United States Government isauthorized to reproduce and distribute reprints for governmental purposesnotwithstanding any copyright notation hereon.

ADMISSIBLE BAYES TESTS FOR STRUCTURALRELATIONSHIP

BY WEI-HSIUNG SHEN

University of Maryland Baltimore County

AND

BIMAL K. SINHA*

University of Maryland Baltimore Countyand

Center for Multivariate Analysis, University of Pittsburgh

Abstract

It is an open problem to construct a test for structural relationship

among the mean vectors of several multivariate normal populations with

known but unequal covariance matrices. In this paper, a class of admissible

Bayes tests for the above problem is derived. As a byproduct, in the special

case of known and equal covariance matrices, the likelihood ratio test ofRao(1iJ73) is shown to be admissible Bayes.

*Research supported by the Air Force Office of Scientific Research (AFOSR) unlder

contract F49620-85-C-008. The United States Governneut is authorized to reproduce anddistribute reprints for governmeutal purposes.

AMS 1980 subject classification. PRIMARY 62H 15; SECONDARY 62('15, (32F15.

Key words and phrases. Structural relationship, admissible'Bayes test , likelihood

ratio test, prior distribution.

1. Introduction. Fisher(1939) pointed out the importance of tests

for structural relationship among the mean vectors of several multivariate

normal populations. To be specific, consider k independent p-variate multi-

normal populations, Np(jui, _,), i = 1,... ,k and the problem of testing

(1.1) H0 Ht = , = l,...,k versus

H, jL 1 , ',Wk arbitrary

where H : s x p is an unknown matrix of known rank s - p and is an

unknown s-vector. The hypothesis H0, if true, implies that the k mean vec-

tors jil, • • -, pk lie in a (p - s)-dimensional subspace of the Euclidean p-space

E P rather than in E' itself, and provides a structural relationship among

the mean vectors. As Fisher(1939) and Rao(1973) rightly emphasized. it is

essential that in dealing with several multinormal populations a hypothesis

of the form (1.1) be tested prior to using the models for prediction etc.

It is an open problem in the literature to construct a suitable test for the

hypotheses in (1.1) based on independent samples from the k populations

when F1, ', Ek are unequal. In the case when EI,' - , V.k are equal (and =

E, say), Rao(1973) derived the likelihood ratio test (LRT) of H,, against 11,

.mmnn~mmm m I

which rejects the null hypothesis for large values of T - A,(BE '). Here

B is the matrix of the sums of squares and products due to the hypotheses

(between groups), A, < A2 < ... < Ap are the ordered roots of BE - ', and

the natrix E is assumed known. If E is unknown, one replaces E in the

detitition of T by 14', the within matrix sum of squares and products due

to error (within groups). That the resultant test is the LRT is proved in

Fujikoshi(1974) and also in Rao(1985).

It is the object of this paper to provide a simple solution to the above

open problem when 11,.... ! are assurned known. We have derived in

Section 2 a class of adnissible Hayes tests for the hypothesis H,, versus

H1. Interestingly enough, inspite of the somewhat complicated structures

of the model and the problem. the derived tests are of extremely simple

form. In the special case of equal and known covariance matrices, the LRT

of Rao(1973) is shown to be admissible Bayes.

We may recall that a Bayes critical region (for 0 1 loss function) is of

the form

(1.2) {x : f(x; 0)r1 (d1) f f(x; )7ro(1)

for some positive constant c, where f(x; 0) is the underlying joint density.

2

0 is the vector of parameters, 7r, and ro are the prior probability measures

over the alternative parameter space E1 and the null parameter space 0o

respectively, and f is over the respective parameter spaces (Kiefer and

Schwartz, 1965). Assuming that we have available a random sample of size

n, from the ith population, denoted as x 1,,.,x,,,, I L ...,k, we can

write

k k

(1.3) f(x;0) = (27r)-p/ 2 -I" , ' 2etr{ -- I 2 n, E1 (l , -

k

•(x, - ,)'}~etr{ I 2 , ',,}

,,here x, - x,3 , S, x,)(x, - x,)', k ,-....,k.

k

n n,, 0 (A 1,, , k), 0 : {( ,... k) E". 1,... k}.1=l

E),,p) Hp, - c -1, tk. 1 s p unknown,

rank(H) = s < p, s known, : s I E"' unknown}, 0, E) -- R,.

k k

In what follows the factor (27r)-"- 2f IV n,-'2etr{ -1'2Z ', 1 } ap-

pearing in f(x;06) is ignored because it is independent of 0 and has n~o

influence on Bayes tests of the form (1.2). 'rhus one can write (1.2) in the

form

(f.) {x f f *(x,)ir (dO), f'(xO) 7,r(dO) -}

3

where

k

(1.5) f'(x;0) = etr{-1I/2 n,t'(x, - -,)(k, -

Appropriate choices of 7r, and 7rT are made in Section 2 and the resultant

Bayes tests are derived.

2. Admissible Bayes tests of (1.1). Under the alternative H 1 , we

choose 7r, as the absolutly continuous measure on the space E P' of the p,'s

given by

(2.1) dir,(pi,. .P ) dg, ... dPk

k k k

(2)-"T kF" I [ 4'-_1 '2 ( ,)P "etr{ '2 V , nA '(i, 0'}xj 2~l t=1

where A, p - p, p.d. matrix, and , p -1 vector, - 1,..., k. Clearly

7r, corresponds to choosing independent normal priors for each t,. The

matrices A, and the vectors ,, = 1.... ,k, will be chosen later. This

immediately results in the numerator of the expression in (1.4) as

(2.2) f f*(x; )rI(dO) =

k k k

(27r) -pk/2(I n,)P"2 J- IA,I -' 2etr{ - 1'2 y_ n,.i, 1x ,x'IZ~I i1 2:=1

k

.etr{ 1,/2 YA,, ,t=-I

k

•etr{1/2Fn,(Et'x, 4 ')( - , + A - ,'t + At )

t=1/kSetr{-1/21n,(

t ± A 1t)(p, - ( +t- 1 + At -

t:=

• (E,-'X, +-A,-',))(p, - (El' - A-')-'(iE-Ix- A-' ,))'}

• d~l ... d~tk

k k- I !.'A + I -1/2 etr{ - 2 n2 A7

1 c }

k

.etr{- 1/21- n> , ,X}t=1

k

• etr{ 2>3n,(Z., + A.'(,) (E IX, + A-' 1)'(<,-t- A-') - '}t=1

The crux of the problem now is to evaluate the denominator of the

expression in (1.4), namely f f°(x:O)7r,(dO) for suitable 7r,'s. Towards this

end, first note that

(2.3) Hp, =

pt = H'(H H') 1 (1,, - H'(H H') -'H)??,

for arbitrary r,- E"

so that 0( can be rewritten as

(2.4) 0() {(PI. . .,.k) A, : H'(IIII') ' + (I - H'(HH') 'H )?,,

7h, EP, I 1 .. ,k, C, T, ... ,rk all arbitrary,

• . i i i5

If , p unknon. rank(1) .s p.., known}.

Our choice of mr) oil (-) corresponds to choosing suitable priors for H

{H ", - p rank(tt) ;*t. c' E r and ',E . 1 .... .k. Throughout

the following. its far as I1 is concvrned, we assume that assigns all its

measure to the .tib ci 4 It ,) I (I. :0) of 4 . It, remains to

specify prior ,;f r ?i1 '7. fl, This is done below

Choose the cold Itoll (le it dc f i . griven . as

k

(2.5) (ll . Il c) 1, 1111k (2,7) kr 2J-J (1 ,) -, "I 1

k0 tr{ ?1X'~ , ' tj, , -k C))(t rl , ' ,)'

I

and the marginal density of c as

(2.6) dr,,() dc (2,7) Q c tr{ I - (C C)( }

where rl," : p I, . 1. p .s. 1,: p p, p.d.. Q :. .sp.d. are to

be suitably chosen. Let

kk

k k

* = (I, :0): .s ,

6

0 0(2.7) pz : P(

0 11,..

k'P 3 = P l( n, E, V),, s s.

'4, - '2EZ, T2 p p. , " k,

, = lqI • p k . - 1....

Then the denominator of the expression in (L.4) can be simplified as

(2.8) f f"(xo)7o(dO)

k k

(27r) -pk 2 n ,)J' 2 J- J)P, 1 (2-,) 2Q ' 1 2k

•etr{- 1 2 >n,, x}ir{ 1 2 ' I r I2Q

k

*etr{- 1, 2Znr1,

f r 12(4'3 Q( Q )

• 4 t, - l ki +-Q -,') ') .i :( , , ,

*etrl, 2> , + 1'P /2 ( P

k

• et,'j -- 1/2 [( +j A, c,7 :" A* , cc ,)lI I

•dcdrjj ... drlk

k k

(2 7) pk p(jj)~ 2 1 1 '-(2 7r) Q ' 2

t-

k

*etr - 1 2NnY ')xetr{ 1- 2Q ' c')Fl"}

k

*etr{ - 1 2 > , n1 I' r~rj:}

et r{ 1 2 n, IV A P,-'0)(\ 3 P, '?7)'

(P14, -P-'e t r{f1 2rTr'= '

etr f - 1 2( q 2 3t t

k

17 , '4t - Ipj- 2 er{-I 2cQ ' r- r~

k k

-tr{ I 2V ,: '.x,tetr{ 1I >'nJ';r"er I 2T-'=

k

.ctr{1 2..,n(42I P, ''7)1'3 'P ' ' 4, P, ) }

wVhere

k

(2.9) B P ,.t '

k

and

k_ I)

(2.10) -= ,i,) + T- En, A1P.-1

k

In the above A,'s, P,'s, and Q are chosen so that _ is p.d.. Hence.

combining (2.2) and (2.8), and taking the logarithm of the ratio of the

expressions in (2.2) and (2.8), the Bayes test turns out to be of the form

k(2.11) t, - n, (E,- :k, + A,- 1 ,(E._1- 1x, + A, 1 ,'': ' .A,- 1'

t=1

k

IC

I:~j n (4' 2 13, - P 'OP)(4., - P y)(r, '71,

SC .

We now choose P,, Q. A. A,. ",. 1,,f and C, suitably subject to - being

p.d. to reduce (2.11) to a nice forn.

Let E, be partitioned as

(2.12) '

where E,(11) : S " ,Y:.,(22) : (p -s) (p - s), and let EI, 1 be similarly

9

partitioned into

( 2 .1 3 ) 1 (N- t 1 t

X 2-1 ,V2 I v 2

i( 1(22 I

where E,(ll 21 an

~k.

(Choose

f) ,ll 0(1) 1. I k,

- I

where t ,, ari , .4 p.d. rratr?.:

k 1 II( 1 1 ) ( " " :, ,

I I

(zi) .4, 1. .k:

(i, At t' , 1 , E ... k"

k

0,t) A," %V PtI, k.tI

and (vii) r7,, t . k, arbitrfra .

10

,,, m~~iu m lia I s hI.lll---III H H•

It may be noted that E simplies to

k

(2.14) --_ 4- Q + n, + > ,Z,,xP5,42Z1

k k

+ '' j + )1 2 .P,+ , ,.-2)=z=1

k k

i=1 2=1

k

t-I

k

2 n1 1

which is clearly p.d. and P,111) does not affect the calculation.

The three terms appearing in the left hand side of (2.11) can now be

evaluated. Combining the first term and the second term of (2.11). and

using ('z). we have

k

(2.15 ) tr{Z n,_" ' -- . )-' - '1 (\4 +),

k

1t

k

t r EnXx:: 4(E +)4I '(P, Jl

2 I1

k

k

I1

kni 0 -+ -(1 ,) '- P 1), , i( 'tA ' I A- -,,P, , + P, ).

t= 1

Since the second term of the last equation does not contain data, we may

drop it. So (2.15) is now

k

(2.16) tr{ 4 in(iqY2)( x)'((Z.' ( A ) 1 +[.2('.'4, + P,-I) 412)}:,=1

k

- tr {1 2 ( x ,(,i ,'

n, (E,l 1 0(00E

(2.17) tr 1r -'- )

k

=tr{1(Iis3)t( 'I,3)(>, n.'X,' ) 0

,=k 0 0

( tr j 112)33-

kn

=tr {1/2(> fl:~X)(*, .fl)'( n'- )' ( Z 1 n2'

tr 1

(} 0

12

Clearly P,(ll) does not affect the above calculations. Combining (2.16) and

(2.17), (2.11) is now of the form

(2.18) tr n,(',- 1 x)' ( )JI,11 0 0

k k ( 1 2)) 0tr (Y- , V ,- 1,k,)(2 nit= -E''0 0

SC.

This immediately prove the following result.

THEOREM 2.1. An admissible Bayes test of (l.1) rejects H. for large

values of the test statistc given by the left hand side of (2.18).

When El, .. , Ek are equal (and E, say), (2.18) reduces to

(2.19) tr jI(nn: n- n n,x,)( n,x,)) E-1t~ ~ l :10 2 i

C.

which is the same as 0 0o(2.20) tr B ( - I

where B is defined earlier.

This leads to the following important result.

THEOREM 2.2. An admissible Bayes test of (1.1) when E1 ...

SE : E rejects H,) for large values of the test statist given by the left

hand side of (2.19) or (2.20).

REMARK 2.1. It is interesting to observe that the test statistic given

by the left hand side of (2.19) can be interpreted as the sum of the first S

roots of the matrix B'-" '. This follows hecause one can alwavs take E It

due to the nature of the hypotheses (1.1) and change 13 to 2 1

REMARK 2.2. It is clear from the preceding calculations that if. under

H,,rno is chosen as assigning all its measure to the sunbspace 4 . . of

the form 1, .. {H :s . p 11 ( 1... ,O... 1, -1, .- 0 ,1,,

(0... I 0- ' :s • 1 with I at the ,th position, I . i, ..... , p., and

appropriate changes are made accordingly in the definitions of kli. %P.,, and

also in the choice of P, Q. A, etc., in the above priors of c and 1j.. rk.

then the Bayes test. rejects H,, for large values of the test statistic

0 0k

(2.21) t'x,)('i4'',)' 0 )

0 0 (

14

* i1 j s, and 1. as the i.th column of H,1.

Ij

o 0 0k k-

tr (n 1 i<1 )(Ej n,t '5c)' 0 kV' I E- I

0 0 0

where E,(i1 2) is the appropriate s , s proper submatrix of E, corresponding

to the rows (i1. ".i,) and the columns (i6. - -, . In the special case

of the equality of El,'" ,Ek, it therefore follows from (2.21) that a test

which rejects It, for large values of the sum of any .; roots of l IE is

admissible Baves. In particular, the likelihood ratio test of Rao(1973) for

equal covariance matrices which rejects H, for large values of the sum of'

smallest roots of BE ' is admissiN ll(B ayes.

15

t EFEIt Nt ES

FISHER. R. A. (1939). The sampi ng (I istri b)ution of some statistics obta Ined

from nonlinear equations. Annals, of Eugemcs, 9, 238-2-19.

FUt IKOSIti, Y. (197-4). The likelihood ratio tests for the dimensionality of

regression coefficients. J. Muit. A nal.. 4. 327-3-10.

KIEFER , .1. and SHIWAHTZ. 1? (1965). Admissible ,ayes character of

T2 , R, , and other fully invariant tests for classical multivariate

normal problems.. rm. .Math. Statist.. 36, 747-770.

R A0). C. R. (197:3). Lnear ,'tatistical biffrence and Its .. pphcat is. 21d.

edition. Wiley. New York.

RAO. C. R. (1985). Tests for dimensionality and interactions of mean

vectors under general and reducib~le covariance structures.

J. Mult. Anal., 16, 173-1S4.

DEPARTMENT OF MATHEMAT.'S

VNIVERSITY OF MARYLANI) BAITIM iRE ( '(NTYCATONSVILLE. MARYLANI) 2 1228

CENTER FOR NIULTIVARIATE ANALY1,SIS

UNIVERSITY OF PITTSBURGI

PITTSBURGH, PENNSYLVANIA 1526(