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4 314 ZHIDONG BAI AND HEWA SARANADASA Proof. Uder the ull hypothesis, by the Cetral Limit Theorem, s (1 ; y) 3 T y ; y!n(0 1) as!1 1 ; y from which the result follows immediately. Now, we cosider the behavior of T = uder H 1. I this case, its distributio is the same as (w + ;1= ) 0 U ;1 (w + ;1= ) (:) where = ; 1 (1 ; ) U= P u iu 0 i w =(w 1 ::: w p ) 0 ad u i i =1 ::: are i.i.d. N(0 I p ) radom vectors ad =(N 1 + N )=N 1 N : Deote the spectral decompositio of U ;1 by O 0 diag[d 1 ::: d p ]O with eigevalues d 1 d p > 0. The, (.) becomes (Ow + ;1= kkv) 0 diag[d 1 ::: d p ](Ow + ;1= kkv) (:3) where v = O=kk. Sice U has the Wishart distributio W ( I p ), the orthogoal matrix O has the Haar distributio o the group of all orthogoal p-matrices, ad hece the vector v is uiformly distributed o the uit p-sphere. Note that the coditioal distributio of Ow give O is N (0 I p ), the same as that of w which is idepedet ofo. This shows that Ow is idepedet ofv. Therefore, replacig Ow i (.3) by w does ot chage the joit distributio of Ow, v ad the d i 's. Cosequetly, T has the same distributio as = px (w i +w i v i ;1= kk + ;1 kk v i )d i (:4) where v =(v 1 ::: v p ) 0 is uiformly distributed o the uit sphere of R p ad is idepedet ofw ad the d i 's. Lemma.. Usig the above otatio, we have p P p d i ; y 1;y P p d i! y (1;y) 3 i probability.! 0 ad Proof. Recallig (.4) with = 0 uder the ull hypothesis ad applyig the Cetral Limit Theorem with D = fd 1 ::: d p g give, we have s T P 1 ; y y + y (1 ; y) 3 x p P q ( 1;y p ; y d y i)+ x (1;y) 3 h = E P P p (w i ; 1)d i p P p d i h p ( 1;y y = E P q p ; d y i)+ p P p d i (1;y) 3 p P p d i x D i i + o(1) (:5)

5 EFFECT OF HIGH DIMENSION 315 where is the distributio fuctio of a stadard ormal variable. O the other had, as show i the proof of Lemma.1, the Cetral Limit Theorem implies that the above quatity teds to (x), for all x. Hece, by the type-covergece theorem (see Page 16 of Loeve (1977)), the lemma is proved. Now we are i positio to derive a approximatio of the power fuctio of Hotellig's test. Theorem.1. If y = p! y (0 1), N 1=(N 1 + N )! (0 1) ad kk = o(1) the H () ; ; + s (1 ; y) y where H () is the power fuctio of Hotellig's test. (1 ; )kk! 0 (:6) Remark.1. The usual cosideratio of the alterative hypothesis i limitig theorems is to assume that p kk! a > 0. Uder this additioal assumptio, it follows from (.6) that the limitig power of Hotellig's test is give by (; +((1;y)=y) 1= (1;)a). This formula shows that the limitig power of Hotellig's test is slowly icreasig for y close to 1, as the o-cetral parameter (amely a) icreases. Proof. Write D =(d 1 ::: d ). Usig the facts Ev 1 =1=p, Ev 4 1 =3=[p(p +)] ad Ev 1 v =1=[p(p + )] ad the applyig Lemma., oe easily obtais E E h px w i v i d i ;1= kk i D = hx p (vi ; Evi ) ;1 kk d i i D = ; kk 4 h p(p +) px d i ; p (p +) px d i kk p! 0 i Pr. (:7) px i d i! 0i Pr. (:8) ad px (Ev i ) ;1 kk d i = 1 p kk px d i = y kk p(1 ; y ) (1 + o p( 1 p )): (:9) Thus, by the above ad Lemma.1, we have H () =P p X w i d i y 1 ; y + s y (1 ; y) 3 p ; y kk p(1 ; y ) + o( p 1 )

6 316 ZHIDONG BAI AND HEWA SARANADASA q y h = E P = ; + P p (w i ; 1)d i pp p d i s (1 ; y) y The proof of Theorem.1 is ow complete. 3. Discussio o Dempster's No-Exact Test p ) (1;y) 3 p ; ykk + o( 1 p(1;y) pp p d i D i (1 ; )kk + o(1): (:10) Dempster (1958, 1960) proposed a o-exact test for the hypothesis described i Sectio, with the dimesio of data possibly greater tha the sample degrees of freedom. First, let us briey describe his test. Deote q N = N 1 + N, X 0 = (x 11 x 1 ::: x 1N1 x 1 ::: x N ) ad by H 0 = ( p 1 N N J N ( N J0 1(N 1+N ) N 1, q N ; 1 N J0 (N 1+N ) N ) 0 h 3 ::: h N ) a suitably chose orthogoal matrix, where J d is a d dimesioal colum vector of 1's. Let Y = HX = (y 1 ::: y N ) 0. The, the vectors y 1 ::: y N are idepedet ormal radom vectors with E(y 1 ) = (N 1 1 +N )= p N, E(y )= ;1= ( 1 ; ), E(y j )=0 for 3 j N Cov(y j )= 1 j N. The, Dempster proposed his o-exact sigicace test statistic F = Q =( P N Q i=3 i=), where Q i = yiy 0 i, = N ;. He used the so-called approximatio techique, assumig Q i is approximately distributed as m r, where the parameters m ad r may besolved by the method of momets. The, the distributio of F is approximately F r r. But geerally the parameter r (its explicit form is give i (3.3) below) is ukow. He estimated r by either of the followig two ways. Approach 1: ^r is the solutio of the equatio Approach : ^r is the solutio of the equatio t + w = t = + 1^r 1+ 1 ( ; 1) (3:1) 1 3^r 1 + 1^r 1+ 1 ( ; 1) + + 3^r 1^r 3 (3:) ^r where t = [l( 1 P N i=3 Q i)] ; P N i=3 l Q i, w = ; P 3i<jN l si ij ad ij is the agle betwee the vectors of y i y j, 3 i<j N. Dempster's test is the to reject H 0 if F >F (^r ^r): By elemetary calculus, we have r = (tr()) tr( ) ad m = tr( ) tr : (3:3)

9 EFFECT OF HIGH DIMENSION 319 Now, we begi to costruct our test. Cosider the statistic M =(x 1 ; x ) 0 (x 1 ; x ) ; trs (4:1) where S = 1 A, x 1 x ad A are deed i Sectio. Uder H 0, we have EM =0. If the coditios (a) - (c) are true, it may be proved (see the Appedix) that uder H 0, M Z = p!n(0 1) as!1: (4:) VarM If the uderlyig distributios are ormal as described i Sectio, the uder H 0 wehave M := VarM = (1 + 1 )tr : (4:3) If the uderlyig distributios are ot ormal but satisfy the coditios (a) - (c), oe may show (see the Appedix) that VarM = M(1 + o(1)): (4:4) Hece (4.) is still true if the deomiator of Z is replaced by M. Therefore, to complete the costructio of our test statistic, we eed oly d a ratio-cosistet estimator of tr( )adsubstitute it ito the deomiator of Z. It seems that a atural estimator of tr should be trs. However, ulike the case where p is xed, trs is geerally either ubiased or ratio-cosistet eve uder the ormal assumptio. If S W p ( ),itisroutietoverify that B = trs ; 1 ( +)( ; 1) (trs ) is a ubiased ad ratio-cosistet estimator of tr. Here, it should be oted that trs ; 1 (trs ) 0, by the Cauchy-Schwarz iequality. I the Appedix, we shall prove that B is still a ratio-cosistet estimator of tr uder the Coditios (a) - (c). Replacig tr i (4.3) by the ratio-cosistet estimator B,we obtai our test statistic Z = (x 1 ; x ) 0 (x 1 ; x ) ; trs trs ; ;1 (trs ) r (+1) (+)(;1) = N 1N N 1+N (x 1 ; x ) 0 (x 1 ; x ) ; trs q (+1) B!N(0 1): (4:5) Due to (4.5) the test rejects H 0 if Z > : Regardig the asymptotic power of our ew test, we have the followig theorem.

10 30 ZHIDONG BAI AND HEWA SARANADASA Theorem 4.1. Uder the Coditios i (a) - (c), (1 ; )kk BS () ; ; + p tr! 0: (4:6) Proof. Let z j be the sample mea of z ij, i =1 ::: j j =1 ad let M 0 =(z 1 ; z ) 0 ; 0 ;(z 1 ; z ) ; tr(s ): The, M 0 has the same distributio as M uder H 0. Thus, Var(M 0 )= M(1 + o(1)) ad M 0 = p Var(M 0 )!N(0 1). Note that M = M 0 ; 0 (z 1 ; z )+kk ad by (3.7) Var( 0 (z 1 ; z )) = 0 = o( tr( )): Hece, Var(M 0 )= Var(M )! 1 ad cosequetly Note that (+1) B= Var(M) 0! 1: Hece, p M;kk!N(0 1): Var(M 0 ) Z ; (1 ; )kk p tr( )!N(0 1): This implies that BS () =P H1 (Z > ) = P M ;kk p Var M 0 = ; + > ; which completes the proof of the theorem. 5. Discussios ad Simulatios (1 ; )kk p tr + o(1) (1 ; )kk p + o(1) (4:7) tr Comparig Theorems.1, 3.1 ad 4.1, we d that from the poit of view of large sample theory, Hotellig's test is less powerful tha the other two tests, whe y is close to oe, ad that the latter two tests have the same asymptotic power fuctio. Our simulatio results show that eve for moderate sample ad dimesio sizes, Hotellig's test is still less powerful tha the other two tests whe the uderlyig covariace structure is reasoably regular (i.e., the structure of does ot cause a too large dierece betwee 0 ;1 ad p kk = p tr( )), whereas the Type I error does ot chage much i the latter two tests. It would ot be hard to see that usig the approach of this paper, oe may easily derive similar results for the oe-sample problem, amely, Hotellig's test

13 EFFECT OF HIGH DIMENSION 33 Table 5.1. Simulated power fuctios of the three tests with multivariate Gamma distributio. N =45 p=40 = :05 =0 =1 = :3 =3:4 = :6 =15:6 = :9 = 35:8 H D BS H D BS H D BS H D BS Table 5.. Simulated power fuctios of the three tests with multivariate ormal distributio. N =45 p=40 = :05 N =45 p=40 = :05 =0 =1 = :5 =41 =0 =1 = :5 =5 H D BS H D BS H D BS H D BS H: Hotellig's F test, D: Dempster's o exact F test, BS: Proposed ormal test, = kp 1; k ad = max tr mi.

14 34 ZHIDONG BAI AND HEWA SARANADASA (p =40 =0:0) (p =40 =0:3) Power Power Power Power (p =40 =0:6) (p =4 =0:9) Hotellig's test ;;;Dempster's test { BS's test Figure 5.1. Simulated power fuctios of the three tests with multivariate Gamma distributio (p =4 =0:0) (p =4 =0:5) Power Power (p =4 =0:0) (p =4 =0:5) Power Power Figure 5.. Simulated power fuctios of the three tests with multivariate ormal distributio. 1.0

15 EFFECT OF HIGH DIMENSION 35 Appedix. Asymptotics Related to the Statistic M A.1. The proof of (4.4): By deitio, we have M =(1+N 1 ;1 )kx 1 k +(1+N ;1 )kx k ; x 0 1 x ; ;1 X XNj j=1 kx ij k : Uder H 0, we may assume 1 = = 0. Write ; = [; 1 ::: ; p ] 0 = [ k `] ad ; 0 ;=[ k`]. The, by Coditios (a) - (c), we have Var( ;1 X XNj j=1 h = ; N tr( )+ Similarly, we may show that kx ij k )= ; E mx `=1 `` i X XNj px j=1 k=1 [(; 0 kz ij ) ;k; k k ] C ;1 [tr + max tr] = o( M): XN 1 Var(x 0 x 1 )=N ; N ; 1 E XN `=1 x 0 i1 x` 1 = tr( ) N 1 N Var(kx 1 k ) = N 1 P tr( )+ N m`=1 ``, P Var(kx 1 3 k = N tr( )+ N m`=1 `` 3 Cov(kx 1 k kx k ) = 0 ad Cov(x 0 1x kx j k )=0forj =1. Therefore, by the fact that P m `=1 `` p max, wehave 1 Var(M )= tr( )+ N1 3 The proof of (4.4) is the complete. + 1 h m i N X`=1 `` = M(1 + o(1)): 3 A.. The asymptotic ormality ofz uder H 0 : From the proof of A.1, oe ca see that (tr(s ) ; tr())= M! 0. Therefore, to show that Z!N(0 1), we eed oly show that[kx 1 ; x k ; E(kx 1 ; x k )]= M!N(0 1). We may rewrite kx 1 ; x k ; E(kx 1 ; x k )= := px k=1 mx `=1 h=1 [(; 0 k(z 1 ; z )) ;k; k k ] NX [U` h + V` h + ``(w ` h ; E(w ` h))]

16 36 ZHIDONG BAI AND HEWA SARANADASA P where z j = N j ;1 N j z ij, z jk deotes the kth compoet ofz j ad h;1 X i U` h = M hw` h ;1 `` w` k1 k 1=1 X `;1 V` h = M ;1 w` h `1=1 `1`(z 1`1 ; z `1) with the covetio that P 0 `1=1 = 0 ad the otatio w` h = 8 >< >: 1 N 1 z h 1 ` if h =1 ::: N 1, 1 N z h;n1 ` if h = N 1 +1 ::: N. Sice Var( P m P N `=1 h=1(w ` h ; E(w ` h))) = ( + )( 1 N N ) P m ``= 1 3 `=1 M! 0, we eed oly show that P m P N `=1 h=1[u` h + V` h ]!N(0 1). Note that fun(`;1)+k = U` k+v` k g forms a sequece of martigale diereces with -elds F N(`;1)+h = F(z ijt j = 1 t < ` i = 1 ::: N j ad w` i i h). The the asymptotic ormality may be proved by employig Corollary 3.1 i Hall (1980) with routie vericatio of the followig: ad m Var X`=1 mx NX `=1 h=1 NX h=1 The proof of (4.) is ow complete. E(U 4` h + V 4 ` h)! 0 E[(U ` h + V` h)jf N(`;1)+h ]! 0: A.3. The ratio-cosistecy of B : We oly eed show that ~ B = trs ; 1 (trs ) is ratio-cosistet for tr( ). Without loss of geerality, we assume that 1 = =0. Note that h X XNj S = ;1 j=1 x ij x 0 ij ; N 1x 1 x 0 1 ; N x x 0 Sice Ex 0 j x j = N ;1 j tr() = o( p tr( )), j = 1, it follows that, x 0 j x j = o( p tr( )). Therefore, we eed oly show that 1 ^B =tr X XNj j=1 x ij x 0 ij 1 ; tr( 1 X XNj j=1 i : x ij x 0 ij)

17 EFFECT OF HIGH DIMENSION 37 is a ratio-cosistet estimator of tr( ). P By elemetary calculatio, we have E(tr( 1 P N j j=1 x ij x 0 ij)) = N tr() ad Var(tr( p P 1 P N j j=1 x ij x 0 ij)) = O(tr( )). These, together with p ;1= tr() = o( tr( )), imply that Rewrite 1 tr 1 tr( 1 X XNj j=1 X XNj j=1 x ij x 0 ij = N tr( )+ N + 1 X X XNj N j 0 X j=1 j 0 =1 i 0 =1 x ij x 0 N ij) = (tr()) + o p (tr( )): X XNj j=1 := N tr( )+H 1 + H : We have E(H 1 )=0adVar(H 1 )= 4N 3 4 Thus, tr((; 0 ;) (z ij z 0 ij ; I m)) (tr((; 0 ;)(z ij z 0 ij ; I m ))(; 0 ;)(z i 0 j0z0 i 0 j ; I m)) 0 h tr( 4 )+ P i m ([(; 0 ;) ] ii = o(tr ( )). H 1 = o p (tr( )): Write H = H 1 + H + H 3 + H 4 + H 5, where H 1 = 1 H = 1 X (ij)6=(i 0 j 0 ) X XNj (tr((; 0 ;)(z ij z 0 ij ; I m))(; 0 ;)(z i 0 j0z0 i 0 j ; I m)) 0 X j=1 (k 0 6=` `06=k) k ` k 0 `0(z ij`z ijk 0z ij`0z ijk ) ad H 3 = H 4 = H 5 = 1 X XNj X j=1 X XNj X `6=k6=`0 j=1 `6=k X XNj mx j=1 k `=1 k `` `0((z ij` ; 1)(z ij`0z ijk )) k `` `((z ij` ; 1)(z ij`z ijk )) k `(z ij` ; 1)(z ijk ; 1):

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