On the Power Function of the Likelihood Ratio Test for MANOVA

Transcription

1 Journa of Mutvarate Anayss 8, (00) do: /jmva On the Power Functon of the Lkehood Rato Test for MANOVA Dua Kumar Bhaumk Unversty of South Aabama and Unversty of Inos at Chcago and Sanat K. Sarkar Tempe Unversty Receved October 7, 1999; pubshed onne January 11, 00 We prove that the power functon of the kehood rato test for MANOVA attans ts mnmum when the rank of the ocaton parameter matrx G decreases from s to 1. Ths provdes a theoretca justfcaton of a resut that s known n the terature based ony on numerca studes. 00 Esever Scence AMS 1980 subject cassfcatons: 6H10; 6H15. Key words and phrases: Wshart dstrbuton; maxma nvarant; zona poynoma; product of beta varates X/0 $ Esever Scence A rghts reserved. 1. INTRODUCTION Suppose X: r p s a random matrx whose rows are ndependenty dstrbuted as mutvarate norma wth common covarance matrx S:p p, and et E(X)=G. Let the random matrx S: p p be ndependent of X and have the Wshart dstrbuton W p (S,n), where S s postve defnte and n \ p. Consder the probem of testng the hypothess H 0 : G=O aganst H 1 : G ] O. (1) Ths s the canonca form of the MANOVA probem. Ths testng probem remans nvarant under a transformatons of the type (X, S) Q (CXPŒ, PSPŒ), () where C:r r s an orthogona matrx, and P: p p s a nonsnguar rea matrx. Let a \ a \ \ a t >0 be the ordered t=mn(r, p) argest egenvaues of XS 1 XŒ and d 1 \ d \ d s >0 be those of W=GS 1 GŒ, 416

2 LIKELIHOOD RATIO TEST FOR MANOVA 417 where s=rank G. A maxma nvarant statstc under ths group of transformatons s (a 1,a,...,a t ), wth (d 1, d,...,d s ) beng a maxma nvarant parameter. The four we-known tests for the above testng probem wth ther acceptance regons are gven beow. 1. Roy s maxmum root test:. Lawey Hoteng trace test: t K =3 (X, S): C K 1 ={(X, S): a 1 [ k 1 }, k 1 >0, a [ k 4,k >0, 3. Lkehood rato test (LRT): t K 3 =3 (X, S): D (1+a ) [ k 3 4,k3 >1, 4. Pa s trace test: t K 4 =3 (X, S): C a /(1+a ) [ k 4 4,0<k4 <1. A number of theoretca resuts expanng how the power functons of these four tests behave wth respect to the non-centraty parameters d 1,...,d s have been derved n the terature. Perman and Okn (1980) proved that any test wth ncreasng rejecton regon n the space of a 1,...,a t,.e., a regon of the form {g(a 1,...,a t ) \ k}, where g s nondecreasng n each argument, s unbased, from whch t foows that a of these four tests are unbased. Let the power functon of a test wth the acceptance regon K be defned as foows p K (D,r,n,p)=P G, S {(X, S) K c }, (3) where K c s the compement of K and D=dag(d 1,...,d s ). Das Gupta et a. (1964) proved that f K s convex n each row of X when S and the remanng rows of X are fxed and f t remans nvarant under a the transformatons defned n (), then the correspondng power functon s ncreasng n each component d,,...,s. Eaton and Perman (1974) notced that the acceptance regons K 1 of Roy s maxmum root test and K of the Lawey Hoteng trace test are convex n (X, S), and they reman nvarant under a the transformatons mentoned n (). Usng a resut of Mudhokar (1966), they proved that for an nvarant convex acceptance regon of ths type, the power functon s Schur-convex n ( `d 1,...,`d s ). Thus, for fxed ; s u=1 `d, the power functons of Roy s maxmum root

3 418 BHAUMIK AND SARKAR test and the Lawey Hoteng trace test ncrease as ( `d 1,...,`d s ) ncreases wth respect to the majorzaton parta orderng. The same monotoncty resut, however, cannot be estabshed for the kehood rato test or Pa s trace test usng the resut of Eaton and Perman (1974) as nether K 3 nor K 4 s a convex set n (X, S). Das Gupta and Perman (1973) proved that the power of the kehood rato test strcty decreases wth p and s. What s conjectured about these tests n the terature based on numerca studes (Fujkosh, 1970) s that for fxed trd=; s d ther power functons decrease as the rank of G decreases from s to 1. Ths artce provdes a parta theoretca support to ths conjecture for the kehood rato test. More specfcay, we prove theoretcay that the power functon of ths test, when ; s d s constant, attans ts mnmum when the rank of D s 1.. THE POWER FUNCTION OF THE LIKELIHOOD RATIO TEST Let us denote the kehood rato test statstc by U and the ower 100a % pont of the nu dstrbuton of U by U a (p,r,n). Let C o (D) denote the zona poynoma of D for the partton o=(k 1,...,k s ), k 1 \ \ k s \ 0, of the nteger k. Let Z s be mutuay ndependent random varabes, where,,...,s, Z = Beta5 n +1, r/+k 6 Beta5 n +1, r/6, =s+1,..., p. We denote the product < p Z by U p, o (r, n) and wrte U p, 0 (r, n) smpy as U p (r, n). Sarkar (1984) has proved that U E K U p, K (r, n), (4) where the expectaton s taken wth respect to K havng the foowng probabty mass functon at o P(o, D)=e C o(d/) trd/. (5) k! Let V s (r, n)=< s Z g, where condtonay gven K=o,Z g s are ndependenty dstrbuted as Beta[ n+r +1,k ],,..., s respectvey. It s assumed that Beta[n, 0]=1 wth probabty one. As defned before o=(k 1,k,...,k s ), where k 1 \ k \ k s \ 0, and k=; s k. If a the k s are not strcty postve then there exsts a postve nteger (1 [ <s),

4 LIKELIHOOD RATIO TEST FOR MANOVA 419 such that k 1 \ k \ \ k >0 and k +1 = =k s =0. From a property of the beta dstrbuton (Rao, 1973, p. 168), t foows that U V s (r, n) U p (r, n). (6) Now we w defne a sequence of random varabes Y g s from Z g s to construct a new random varabe V g wth the property that t s stochastcay arger than V s. To ths end, et us consder Y g Beta1 n+r +k +1, C j=1 k j,,,..., 1, (7) and then defne V g =Y g 1Z g,,...,, wth Y g 0 =1. In the foowng, the nequates between random varabes are referred to as stochastc nequates. Note that V g =Beta1 n+r +1, C k j j=1 k j [ Beta1 n+r +k +1, C =Y g,,..., 1. The second nequaty above hods from the fact that k +1 \ 1, = 1,,..., 1, and that a beta random varabe Beta(p, q) stochastcay ncreases wth p. Thus we have s V s =D Z g =D Z g j=1 [ Y g 1 D Z g =V g D Z g = =3 [ Y g D Z g =V g 3 D Z g =3 =4 [ [ V g. (8) In other words, we have V g n+r +1 Beta(, ; n+r +1 k )=Beta(,k) whch s stochastcay arger than V s (r, n). Aso note that V g s stochastcay smaer than V 1 Beta( n+r,k) as \ 1. Usng ; o C o (D)=(trD) k we

5 40 BHAUMIK AND SARKAR now see that the power functon, say p s (D), of the kehood rato test for MANOVA satsfes the foowng p s (D)=P(U [ U a (p,r,n)) =P(V s U p (r, n) [ U a (p,r,n)) \ P(V g U p (r, n) [ U a (p,r,n)) \ P(V 1 U p (r, n) [ U a (p,r,n)). = C k=0 C o e C o(d/) trd/ P(V k! 1 U p (r, n) [ U a (p,r,n)). trd/ (trd/)k = C e P(V k=0 k! 1 U p (r, n) [ U a (p,r,n)) =p 1 (trd). (9) Thus, the power functon of the kehood rato test for MANOVA, for fxed trd, attans ts mnmum at s=1. Ths proves our man resut. Remark 1. From a computatona pont of vew, the extreme rghthand sde of (9) s very usefu. It provdes a frst-hand approxmaton to the power functon of the kehood rato test for MANOVA whch, beng based on the Posson dstrbuton as opposed to the more compcated dstrbuton nvovng zona poynomas, coud be computed reatvey easy. It s mportant to pont out that n (9) we have ony proved that the power of the kehood rato test for MANOVA at D=(d 1,...,d s ) s more than that at D 0 =(trd,0,...,0). Athough D 0 majorzes D, the current resut cannot be generazed to make a cam that whenever any D s majorzed by D 0, the power of the kehood rato test for MANOVA at D s more than D 0, or n other words, the power functon s Schur-concave. Aso, the current technque does not work to study the smar property of the power functon of Pa s trace test. ACKNOWLEDGMENTS The authors are gratefu to two referees and an edtor for ther usefu comments and suggestons.

6 LIKELIHOOD RATIO TEST FOR MANOVA 41 REFERENCES 1. S. Das Gupta, T. W. Anderson, and G. S. Mudhokar, Monotoncty of the power functons of some tests of the mutvarate near hypothess, Ann. Math. Statst. 35 (1964), S. Das Gupta, and M. D. Perman, On the power of Wks U-test for MANOVA, J. Mutvarate Ana. 3 (1973), M. L. Eaton, and M. D. Perman, A monotoncty property of the power functons of some nvarant tests for MANOVA, Ann. Statst. 5 (1974), Y. Fujkosh, Asymptotc expansons of the dstrbutons of test statstcs n mutvarate statstcs, J. Sc. Hroshma Unv. Ser. A-I. 34 (1970), G. S. Mudhokar, The ntegra of an nvarant unmoda functon over an nvarant convex set- an nequaty an appcatons, Proc. Amer. Math. Soc. 17 (1966), R. J. Murhead, Aspects of Mutvarate Statstca Theory, Wey, New York, M. D. Perman, and I. Okn, Unbasedness of nvarant tests for MANOVA and other mutvarate probems, Ann. Statst. 6 (1980), C. R. Rao, Lnear Statstca Inference and Its Appcatons, second ed., Wey, New York, S. K. Sarkar, A note on the power of the kehood rato test for MANOVA, Sankhya, Ser. A 46 (1984),