TESTING FOR ORDER RESTRICTION ON MEAN VECTORS OF MULTIVARIATE NORMAL POPULATIONS

Transcription

1 TESTING FOR ORDER RESTRICTION ON MEAN VECTORS OF MULTIVARIATE NORMAL POPULATIONS Abouzar Bazyar ad Zahra Hedar Departmet of Statstcs, Persa Gulf Uversty, Bushehr, Ira ABSTRACT: Multvarate sotoc regresso theory plays a ey role the feld of testg statstcal hypotheses uder order restrcto for vector valued parameters. Ths d of statstcal hypothess testg has bee studed to some extet, for example, by Kulatuga ad Sasabuch (1984) whe the covarace matrces are ow ad also Sasabuch et al. (003) ad Sasabuch (007) whe the covarace matrces are uow but commo. I the preset paper, we are terested a geeral testg for order restrcto of mea vectors agast all possble alteratves based o a radom sample from several p dmesoal ormal populatos whe the uow covarace matrces are commo. I fact, ths problem of testg s a exteso of Bazyar ad Chpardaz's (01) problem. We propose a test statstc by lelhood rato method based o orthogoal projectos o the closed covex coes, study ts upper tal probablty uder the ull hypothess ad estmate ts crtcal values for dfferet sgfcace levels by usg Mote Carlo smulato. The problem of testg ad obtaed results s llustrated wth a real example where ths ferece problem arses to evaluate the effect of Vyldee fluorde o lver damage. KEYWORDS: Mote Carlo smulato; Multvarate sotoc regresso; Multvarate ormal populato; Testg order restrcto. AMS (000) subject classfcato: Prmary 6F30; secodary 6F03, 6H15. INTRODUCTION Problems cocerg estmato of parameters ad determato the statstc, whe t s ow a pror that some of these parameters are subject to certa order restrctos, are of cosderable terest. There are may szeable lteratures dealg wth meas testg problem uder order restrctos. Bartholomew (1959), cosdered the problem of testg the homogeety of several uvarate ormal meas agast a order restrcted alteratve hypothess. I may applcatos researchers are terested testg for equalty costrats amog populato meas vectors μ, 1,,,, after adjustg for covarates. For stace, toxcologsts are ofte terested studyg the effect of a chemcal o the mea weght of a specfc orga of a amal after adjustg for ts body weght (Kao et al. 00a, 00b). Istead of the usual two-sded alteratve μ μ j j, researchers are ofte terested testg agast equaltes amog the parameters (ow as order restrctos). Some commo order restrctos of terest the multvarate dstrbutos (wth at least oe strct equalty) are; (a) Smple order μ μ j, for j, where ths uequal meas that all the elemets of μ j μ are o-egatve. (b) Smple tree order μ1 μ j, for 1 j, (c) Umbrella order (wth pea at ) 19

2 μ1 μ μ μ 1 μ. The ull hypothess beg H0 :μ1 μ μ (wth at least oe strct equalty). Robertso ad Wegma (1978), obtaed the lelhood rato test statstc for testg the sotocess of several uvarate ormal meas agast all alteratve hypotheses. They calculated ts exact crtcal values at dfferet sgfcace levels for some of the ormal dstrbutos ad smulated the power by Mote Carlo expermet. Also they cosdered the test of tred for a expoetal class of dstrbutos. Sasabuch et al. (1983), exteded Bartholomew's (1959) problem to multvarate ormal mea vectors wth ow covarace matrces. They computed the lelhood rato test statstc ad proposed a teratve algorthm for computg the bvarate sotoc regresso. Sasabuch et al. (003), geeralzed Bartholomew's (1959) problem to that of several multvarate ormal mea vectors wth uow covarace matrces. They proposed a test statstc, studed ts upper tal probablty uder the ull hypothess ad estmated ts crtcal values. Sasabuch (007), provded some tests, whch are more powerful tha Sasabuch et al. (003). Bazyar (01), preseted some propertes of testg homogeety of multvarate ormal mea vectors agast a order restrcto for two cases, the covarace matrces are ow, ad the case that they have a uow scale factor. He computed the crtcal values for the proposed test statstc by Kulatuga ad Sasabuch (1984) for the frst case at dfferet sgfcace levels for some of the two ad three dmesoal ormal dstrbutos. The power ad p value of test statstc are computed usg Mote Carlo smulato. Also whe the covarace matrces have a uow scale factor the specfc codtos are gve whch uder those the estmator of the uow scale factor does ot exst ad the uque test statstc s obtaed. Bazyar ad Chpardaz (01), geeralzed Robertso ad Wegma's (1978) problem to that of several multvarate ormal mea vectors wth uow covarace matrces. They proposed a test statstc, studed ts upper tal probablty uder the ull hypothess ad estmated ts crtcal values usg Mote Carlo smulato. Bazyar ad Pesar (013), cosdered testg the homogeety of mea vectors agast two-sded restrcted alteratves separately multvarate ormal dstrbutos ad examed the problem of testg uder two separate cases. Oe case s that covarace matrces are ow, the other oe s that covarace matrces are uow but commo. I two cases, the test statstcs are proposed, the ull dstrbutos of test statstcs are derved ad ts crtcal values are computed at dfferet sgfcace levels. The power of tests studed va Mote Carlo smulato. Bazyar (016), cosdered testg homogeety of multvarate ormal mea vectors uder a order restrcto whe the covarace matrces are completely uow, arbtrary postve defte ad uequal. The bootstrap test statstc proposed ad because of the ma advatage of the bootstrap test s that t avods the dervato of the complex ull dstrbuto aalytcally ad s easy to mplemet, the bootstrap p value defed ad a algorthm preseted to estmate t. The power of the test estmated for some of the p dmesoal ormal dstrbutos by Mote Carlo smulato. Also, the ull dstrbuto of test statstc evaluated usg erel desty. The problem of estmatg the uow parameter μ, 1,,, p, uder equalty costrats has receved cosderable atteto may boos. For a excellet revew o ths subject oe may refer to the boos by Slvapulle ad Se (005) ad va Eede (006). Suppose that X1,X,,X are radom vectors from a p dmesoal ormal dstrbuto Np μ, wth uow mea vector μ, 1,,,, ad osgular covarace matrx. We assume that s uow. Cosder the problem of testg 0

3 H0 :μ1 μ μ, agast the hypothess H1, where H1 s all possble alteratves o the mea vectors. Stll cosder p dmesoal ormal dstrbutos X ~ Np μ,, 1,,,, where μ (μ (1),μ () ), 1,,,. I geeral, we say that μ (μ1,μ,,μ ) R p, for ay μ ( 1,,, p ) R p, 1,,,, s ordered o colums, or smply μ s o ordered matrx, f μ1 μ μ. Suppose that the dmeso of μ (1) s s r ad the dmeso of μ () s p r. I the preset paper, we are terested the problem of testg H0 :μ1(1) μ(1) μ(1),μ1() μ() μ(), agast all alteratve hypotheses o the mea vectors. Also the preset paper, we suppose that the commo covarace matrces are uow. It s clear that f r 0, ths testg problem s the testg problem gve Bazyar ad Chpardaz (01). Therefore ths testg problem s a exteso of Bazyar ad Chpardaz (01). Such tests may be used some felds. Ths d of testg represetato s commo, for stace, selecto ad rag problem for fdg the largest elemet of several ormal meas (see Shmodara, 000). Sara et al. (1995) ad Slvapulle ad Se (005) dscuss other examples from dfferet areas, especally medce. Also ther applcatos ca be foud clcal trals desg to test superorty of a combato therapy (Lasa ad Meser, 1989 ad Sara et al., 1995). Cosder the followg example. Example 1. A survey s coducted amog the studets 4th grad, 5th grad ad mxed grads dstct I, ad amog the studets 4th grad ad 5th grad dstct II. Observatos o four varables: the age, the household come, the heght ad the umber of hours for oacademc actvtes per wee schools are collected. The meas are represeted as elemets matrx μ R 4 5 ad gve Table 1. Table 1. Structure of the mea vector elemets expermet o the studets 4th grad 5th grad Mxed 4th grad 5th grad Dstct I Dstct I grads Dstct Dstct Dstct I II II Age Icome Heght Play hours 1

4 Oe may assume that the equaltes , , , wth at least oe strct equalty oe of them s establshed. So we have the ordered hypothess H0 : μ1 μ μ3 μ4, wth at least oe strct equalty. The rest of ths paper s orgazed as follows. I Secto, the problem of testg s descrbed, two deftos are gve ad a test statstc s proposed. I Secto 3, the ull dstrbuto of the test, two lemmas ad ma theorem are gve. I Secto 4, the crtcal values of the test statstc whe the sample szes are detcal ad also whe they are dfferet are estmated usg Mote Carlo smulato. The problem of testg s appled to a applcato example Secto 5. Cocludg remars are gve Secto 6. The complete source programs are wrtte softwares PLUS. The problem of testg Cosder p dmesoal ormal dstrbutos X ~ Np μ,, wth observatos Xj, j 1,,, 1,,,. I the preset paper, we are terested testg H0 :μ1(1) μ(1) μ(1),μ1() μ() μ(), agast all alteratve hypotheses o the mea vectors whe the uow covarace matrces are commo. 1 Let X j 1 Xj ad S 1 j 1 (Xj X )(Xj X ) be the sample mea vector of th populato ad sample mea varace covarace matrx respectvely. Defto 1 (Sasabuch et al., 1983). Gve p varate real vectors X1, X,, X ad p p postve defte matrces 1,,,, a p real matrx (μˆ1,μˆ,,μˆ) s sad to be the multvarate sotoc regresso (MIR) of X1, X,, X wth weghts 1 1, 1,, 1, f (μˆ1 μˆ μˆ) ad (μˆ1,μˆ,,μˆ) satsfes

5 μ m μ (X μ) μ (X μ ) (X μˆ ) 1 (X μˆ ), where μˆ ' s ca be computed by the teratve algorthm proposed by Sasabuch et al. (1983). I fact, ths defto cludes the defto gve Barlow et al. (197) for uvarate varables. Defto. c s called a covex coe f x, y c, 0, 0, the x y c. Also c s called a closed covex coe f t s covex coe ad close set. Defe two closed covex coes c0 ad c1 R p by μ1 c0 μ μ1(1) μ(1) μ (1) p, 1,,,,,μ1() μ() μ (),μ R μ μ1 c1 μ μ R p, 1,,,, μ where uder the closed covex coe c1 there s o ay restrcto o the mea vectors μ. Suppose that μˆ, 1,,,, s the MIR of uow parameter μ uder the closed covex coe c0. The we have 1 (X μˆ ) S 1(X μˆ ) mμ c0 1 (X μ ) S 1(X μ ). For p dmesoal real vectors x (x1,x,,x ) ad y (y1,y,,y ) ther er product R p s defed as x,y Λ x Λ 1 y 1 3

6 ) y1 (x1,x,,x 0 1 y Also defe a orm. R p by x x,x Λ. Suppose that for x R p, Λ(x,c) be the pot whch mmzes x w Λ, where w c. We ote that, sce c s a closed covex coe, so the uqueess of Λ(x,c) s clear. Let A B be the Kroecer product of matrces Ar m (aj ) ad Bh s (bl ) ad defed as a11b a1mb A B. ar1b armb Therefore x,y Λ x (D Λ 1 )y, where D The test statstc The lelhood fucto for testg H 0 versus H1 s L(μ1,μ,,μ, ) 1 ( 1) p/ exp 1 (Xj μ ) 1(Xj μ ) j 1 1 exp 1 4

7 ( ) p/ 1 j 1 (Xj μ ) 1(Xj μ ) S, ( 1) p/ exp 1 tr 1 (X μ )(X μ ) 1 dstrbuted wth Wshart dstrbuto Wp (, ) ad. where S s Suppose that A s a p p o-egatve defte real (symmetrc) matrx, 1,,, p are the characterstc roots of A ad s a postve umber, the 1 p p I p A (1 ) 1 O( ) 1 tr( A) O( ). (1) 1 1 By Aderso (1984), t s well ow that the supremum of the fucto L(μ1,μ,,μ, ) o 0 whch s the supremum for all the p p postve defte matrces gve by () p max L(μ1,μ,,μ, ) e S 1 (X μ )(X μ ), 0 Therefore we have H max 0, 0 L(μ1,μ,,μ, ) e S 1 (X μˆ )(X μˆ ), (3) ad also t s completely clear that e max L(μ1,μ,,μ, ) S. (4) p p 5

8 H1, 0 From equatos (1) ad () we get that p max 0 L L(μ1,μ,,μ, ) L e L S (X μ )(X μ ) 1 p ) S L e L S I p S 1 1 (X μ )(X μ 1 ) S p e L S L I p S 1 (X μ )(X μ L 1 p 1 L e p L S L 1 tr S L e L S O the other had L S 1 e L ( X p μ ˆ )( X L S μ ˆ )' L 1 ( X μ p e L 1 tr S 1 (X μ )(X μ ) 1 1 (X μ )(X μ ) 6

9 1 1(X μ ), ) S 1 max L L(μ 1,μ,,μ, ) H0, 0 (5) L S m μ H0 (X μ ) S 1(X μ ) 1 L S (X μˆ ) S 1 (X μˆ ). 1 By equatos gve (3) ad (4), the lelhood rato test statstc s 1 ( X S Therefore S L L S 1 μ ˆ )( X X μˆ S, ( X μ ˆ )( X μ ˆ )'. μ ˆ )' L S. The from equato (5) we have L 1 (X μˆ ) S 1(X μˆ ) where X (X1,X,,X ) ad μˆ (μˆ1,μˆ,,μˆ ). Thus the test statstc by LRT method gve by T X μˆ S. For gve sgfcace level, we reject the ull hypothess H 0, whe T t, where t s a postve costat depedg o the sgfcace level. The ull dstrbuto of the test statstc T To obta the ull dstrbuto of the statstc T, frst we deote by T. The T X μˆ S X S (X,c0) S (6) If H :μ1 μ μ, the H s the least favorable amog hypotheses satsfyg H0 wth the largest type I error probablty (Slvapulle ad Se, 005). Therefore for gve the sgfcace level, we have sup Pμ0, (T t ), where 7

10 0 μ0 s the commo value of μ1,,μ uder H. Now, easly we have the followg theorem. Theorem 1. Uder the hypothess H, the dstrbuto of T gve (6) s depedet of μ0. Proof. Defe the radom vector Y by Y1 X1 μ0 μ0 Y X. Y X μ0 μ0 The t s clear that the dstrbuto of Y depedet of μ0 ad s dstrbuted wth N p (0, ). O the other had T X S (X,c0) S Y S (Y,c0) S. Sce the dstrbuto of S (Y,c0) Y S s depedet of μ0, so the dstrbuto of T statstc s depedet of μ0 ad ths completes the proof. Defe the closed covex coe c as μ1 c μ μ1(1) μ(1) μ(1), 1()1 ()1 ()1, μ where ()1 er 1μ, 1,,, ad er 1 s a p dmesoal vector wth the (r 1) th elemet beg oe, others are zero. Also we defe aother statstc T * X S (X,c0) S X S (X,c) S. Sce the computato of the crtcal values from the formula supsuppμ, (T t ) H0 8

11 s dffcult, we wll show that the dstrbuto of T * s depedet of μ0 ad commo value of μ1,μ,,μ., where μ0 s the If μˆˆ (μˆˆ 1,μˆˆ,,μˆˆ ) s the multvarate sotoc regresso of X1, X,, X uder the closed covex coe c, the T * (X μˆ ) S 1 (X μˆ ) (X μˆˆ ) S 1 (X μˆˆ ). 1 1 Suppose that M s a p p osgular postve defte matrx gve by M11 0 M M 1 M, (7) where M11 s a r r dmeso matrx ad M s a (p r) (p r) dmeso matrx. Also put I 0 D 1 ad DM. 0 M The we get that M11 0 1M 1 I. M Put E M (8) E1 E M 1M 1M11 1 I Lemma 1. For matrx M gve (7), we have a) For ay p p orthogoal matrx H, (I (HM ))c0 c0. b) There exsts a p p orthogoal matrx H whch satsfes: (I (HM))c c. 9

12 Proof. The proof of part (a) s easy to derve. We oly prove the part (b). Put Ir * [Ir 0], the by (8) t s clear that Ir * 1 [M11 1 0]. Let r 1 er 1. The T μ1 (I )c μ1(1) μ(1) μ (1),μ1()1 μ()1 μ ()1 T μ 1 T μ1 Ir* μ Ir* μ( 1),er 1 μ er 1 μ( 1) 1 T μ 1 1 * 1 Ir* 1 ( 1),er 1 1 er 1 1 ( 1) Ir M11 1 (1) M11 1 ( 1)(1),er 1 1 er 1 1 ( 1) (1) ( 1)(1), r 1 r 1 ( 1) (1) ( 1)(1), ()1 ( 1)()1 c. 1 O the other had (I M )c (I (D 1 T)) c (I D 1 )c 1 D 1 μ1 30

13 μ μ(1) μ( 1)(1),μ() μ ( 1)() 1 D 1 1 D 1 μ1 * Ir*μ( 1),er 1 μ er 1 μ ( 1) Ir μ 1 D 1 μ * D Ir* D ( 1),er 1 D er 1 D ( 1) Ir 1 1 * Ir* ( 1),er 1 D er 1 D ( 1). Ir 1 Put a M 1 e1,( p r) 1, where e1,( p r) 1 s a (p r) -dmesoal vector of the form (1,0,,0), ad 11 s the postve umber such that a1 a1 1. There exsts a (p r) (p r) orthogoal matrx H such that H a1 e1,(p r) 1. Put I 0 H 0 H. The for matrx H, we have 31

14 (I (HM))c (I H)(I M)c H 1 1 * Ir* ( 1),er 1 D er 1 D ( 1) Ir 1 H 1 1 * H Ir* H ( 1),er 1 D H er 1 D H ( 1) Ir 1 1 H 1 1 * I *,e M 1 H e M I r r ( 1) 1,( p r) 1 () 1,( p r) 1 ( 1)() * Ir* ( 1), 11 e1,( p r) 1 () 11 e1,( p r) 1 ( 1)() Ir 1 3

15 1 1 * Ir* ( 1), er 1 er 1 ( 1) c, Ir 1 ad ths completes the proof. Lemma. Uder the hypothess H, the dstrbuto of T * s depedet of μ0 ad, where μ0 s the commo value of μ1,μ,,μ. Proof. It s clear that the dstrbuto of T * s depedet of μ0. Put W (X μ0) J, where J s the vector of 1 s. To arbtrary postve defte real matrx, there exsts lower tragular o-sgular matrx M wth postve dagoal elemets satsfyg M M I p. Let H be the orthogoal matrx whch satsfes the part (b) of lemma 1. The T * W S (W,c0 ) S W S (W,c ) S (I (HM ))W HMSM H ((I (HM ))W,(I (HM ))c0 ) HMSM H (I (HM ))W HMSM H ((I (HM ))W,(I (HM ))c ) HMSM H. Put Z1 HMW1 Z HMW * HMSM H. Z (I (HM ))W ad S Z HMW The by lemma 1, we have T * Z S (Z,c0 ) S* Z S (Z,c ) S*. By ther deftos, S * ad Z1,,Z are mutually depedet, S * ad Z dstrbuted as Wp 1 (, I p ) ad N p (0, I p ), 1,,, respectvely. Ths completes the proof. Suppose that 33

16 I r 0 F 0 A, where Ir s the r r detty matrx ad A s a (p r) (p r) osgular matrx defed by A. 0 1 It s clear that f r 0, the F s gve lemma 6 of Bazyar ad Chpardaz (01). Now, we have the followg ma theorem. Theorem. For the real umber t depedg o the sgfcace level, supsup Pμ, (T t ) P0,Ip (T * t ). H0 Proof. It s completely clear that T T *. The by lemma, we get that H 0 (9) supsup Pμ, (T t ) sup P0, (T t ) sup P0, (T * t ) P0,Ip (T * t ). O the other had, we show that supsup Pμ, (T t ) P0,I p (T * t ). H0 Usg the lemmas 7 ad 8 gve Bazyar ad Chpardaz (01), t s easy to show that P0, (T t ) P0,Ip X S (X,c0 ) S, where (F F) 1. Also lmp0, (T t ) lm P0,Ip X S (X,c0 ) S t 34

17 P0,Ip S (X,c ) X S t sce S (X,c0) X S 0. So that P0,Ip T * S (X,c0 ) X S t t P0,Ip (T * t ), supsup Pμ, (T t ) sup P0, (T t ) H0 (10) lm P0, (T t ) P0,Ip (T t ). From (9) ad (10) the proof of theorem s complete. Therefore to compute the crtcal values of the test statstc t s eough to obta that of T * whe μ 0 ad I p. The crtcal values I ths secto, the crtcal values of the test statstc T are estmated by Mote Carlo smulato method. To obta these values, by theorem, we oly eed to obta that of T * whe μ 0 ad I p. I ths smulato, we geerate sets of p varate ormal vectors from Np (0,I) ad compute the statstc T *. Ths computato s repeated tmes to get a estmated upper pot of T *. We further repeat ths process 10 tmes ad compute the average of the 10 estmated upper pot for 0.01,0.05,0.05, (p 3, 4,r 1), (p 4, 5,r ), (p 5, 4,r 3), ad 5,10,15, 0, 5, 1,,,, respectvely. The estmated crtcal values are gve Table. Also the crtcal values of test statstc are estmated whe the sample szes are dfferet. The estmated crtcal values are gve Table

18 Table. Estmated crtcal values of test statstc by smulato whe the sample szes are detcal p r Table 3. Estmated crtcal values of test statstc by smulato whe the sample szes are dfferet p r Crtcal value

19 A example The problem we are cosderg comes from Detz (1989). Vyldee fluorde s suspected of causg lver damage. A expermet was carred out to evaluate ts effects. Four groups of 10 male Fscher-344 rats receved, by halato exposure, oe of several dosages of vyldee fluorde. Amog the respose varables measured o the rats were three serum ezymes: SDH, SGPOT, ad SGPT. It s ow the scetfc cosderatos that the respose level of the ezyme SDH would ot be affected by the dosage levels of vyldee fluorde ad the resposes of the other two ezymes would be affected mootocally. The data are gve Table 4. Let Xj (Xj1, Xj, Xj3) deote the observatos o the three ezymes for j th subject ( j 1,,10) treatmet ( 1,,4). Let deote the mea respose for th treatmet (.e. dose) ad th varable ad let μ ( 1,, 3 ) for 1,,4. Suppose that we defe μ (1) 1 ad μ () (, 3 ). Now, oe formulato of the ull ad alteratve hypothess s 37

20 H0 :μ1(1) μ(1) μ4(1),μ1() μ() μ4(), agast all alteratve hypotheses o the four mea vectors for sgfcace level Table 4. Serum ezyme levels rats Dosage Rat wth dosage Ezyme 0 SDH SGPOT SGPT SDH SGPOT SGPT SDH SGPOT SGPT SDH SGPOT SGPT From the data, we have X The by teratve algorthm to compute multvarate sotoc regresso gve by Sasabuch et al. (199), uder the closed covex coe c0 the estmate of μ s μˆ , 38

21 ad uder the closed covex coe c the estmate of μ s μˆˆ So that X μˆ , ad X μˆˆ The sample mea varace covarace matrx ad ts verse are S , e-004 Also the value of test statstc T * s ad e e e-00 S e e T * 10 (X μˆ ) S 1 (X μˆ ) (X μˆˆ ) S 1 (X μˆˆ )

22 Sce at sgfcace level 0.05, T * 0.667, therefore we reject the ull hypothess. CONCLUDING REMARKS Bazyar ad Chpardaz (01) cosdered the problem of testg order restrcto of mea vectors agast all possble alteratves based o a sample from several p dmesoal ormal dstrbutos. They obtaed a test statstc ad also preseted Mote Carlo smulato to estmate ts crtcal values. I ths artcle, the geeral form for ths problem of testg s cosdered. I fact, ths paper dd umercal study based o the clam that the tal probablty of a proposed test statstc T for testg order restrcted ull hypothess ca be smplfed by aother smpler statstc T *. We proposed a test statstc by lelhood rato method based o orthogoal projectos o the closed covex coes. Mote Carlo smulato s used to obta the crtcal values of test statstc. We also appled ths test to a real example where ths hypothess problem arses to evaluate the effect of Vyldee fluorde o lver damage. For computg the test statstc umercal example the estmato of uow parameter vector s doe by the teratve algorthm proposed by Sasabuch et al. (1983). REFERENCES Aderso, T. W. (1984). A Itroducto to Multvarate Statstcal Aalyss. d edto, New Yor: Joh Wley. Barlow. R. E. Bartholomew, D. J., Bremer, J. M. ad Bru, H. D. (197). Statstcal Iferece uder Order Restrctos: The Theory ad Applcato of Isotoc Regresso. Joh Wley, New Yor. Bartholomew, D. J. (1959). A test of homogeety for ordered alteratves. Bometra, 46, Bazyar, A. (01). O the Computato of Some Propertes of Testg Homogeety of Multvarate Normal Mea Vectors agast a Order Restrcto. METRON, Iteratoal Joural of Statstcs, 70 (1), Bazyar, A. (016). Bootstrap approach to test the homogeety of order restrcted mea vectors whe the covarace matrces are uow, DOI: / , Accepted for Publcato Commucatos Statstcs, Computato ad Smulato. Bazyar, A. ad Chpardaz, R. (01). A test for order restrcto of several multvarate ormal mea vectors agast all alteratves whe the covarace matrces are uow but commo. Joural of Statstcal Theory ad Applcatos, 11(1), Bazyar, A. ad Pesar, F. (013). Parametrc ad permutato testg for multvarate mootoc alteratves. Statstcs ad Computg, 3 (5), Detz, E. J. (1989). Multvarate geeralzatos of Jocheere's test for ordered alteratves. Commucato Statstcs, Theory ad Methods, 18, Kao, J., Oyo, L., Peddada, S. D., Ashby, J., Jacob, E. ad Owes, W. (00a). The OECD program to valdate the uterotrophc boassay: Phase Two-Dose Respose Studes. Evrometal Health Perspectves, 111, Kao, J., Oyo, L., Peddada, S. D., Ashby, J., Jacob, E. ad Owes, W. (00b). The OECD program to valdate the rat uterotrophc boassay: Phase Two-Coded Sgle Dose Studes. Evrometal Health Perspectves, 111,

23 Kulatuga, D. D. S. ad Sasabuch, S. (1984). A test of homogeety of mea vectors agast multvarate sotoc alteratves, Mem Fac Sc, Kyushu Uv Ser A Mathemat, 38, Lasa, E. M. ad Meser, M. J. (1989). Testg whether detfed treatmet s best. Bometrcs, 45, Robertso, T. ad Wegma, E. T. (1978). Lelhood rato tests for order restrctos expoetal famles. The Aals of Statstcs, 6(3), Sara, S. K., Sap, S., ad Wag, W. (1995). O mprovg the m test for the aalyss of combato drug trals (Corr: 1998V60 p ). Joural of Statstcal Computato ad Smulato, 51, Sasabuch S, Iutsua M. ad Kulatuga D. D. S. (199). A algorthm for computg multvarate sotoc regresso. Hroshma Mathematcal Joural,, Sasabuch, S. (007). More powerful tests for homogeety of multvarate ormal mea vectors uder a order restrcto. Sahya, 69(4), Sasabuch, S., Iutsua, M. ad Kulatuga, D. D. S. (1983). A multvarate verso of sotoc regresso. Bometra, 70, Sasabuch, S., Taaa, K. ad Taesh, T. (003). Testg homogeety of multvarate ormal mea vectors uder a order restrcto whe the covarace matrces are commo but uow. The Aals of Statstcs, 31(5), Shmodara, H. (000). Approxmately Ubased Oe-Sded Tests of the Maxmum of Normal Meas Usg Iterated Bootstrap Correctos. Techcal report o , Departmet of Statstcs, Staford Uversty, Staford. Slvapulle, M. J. ad Se, P. K. (005). Costraed Statstcal Iferece: Iequalty, Order, ad Shape Restrctos, Joh Wley, New Yor. va Eede, C. (006). Restrcted Parameter Space Estmato Problems Admssblty ad Mmaxty Propertes. New Yor, USA. 41