SOME TESTS CONCERNING THE COVARIANCE MATRIX IN HIGH DIMENSIONAL DATA

Transcription

1 J. Jaan Statist. Soc. Vol. 35 No SOME TESTS CONCERNING THE COVARIANCE MATRIX IN HIGH DIMENSIONAL DATA Muni S. Srivastava* In this aer, tests are develoed for testing certain hyotheses on the covariance matrix Σ, when the samle size N = n+ is smaller than the dimension of the data. Under the condition that tr Σ i / exists and > 0, as, i =,...,8, tests are develoed for testing the hyotheses that the covariance matrix in a normally distributed data is an identity matrix, a constant time the identity matrix sherecity, and is a diagonal matrix. The asymtotic null and non-null distributions of these test statistics are given. Key words and hrases: Asymtotic distributions, multivariate normal, null and non-null distributions, samle size smaller than the dimension.. Introduction Recent advances in analyzing high dimensional data with fewer observations require that certain assumtions made imlicitly or exlicitly in analyzing them should be ascertained. For examle, Dudoit et al. 00 in their analysis of microarrays data on genes assume that the covariance matrices are diagonal matrices, and thus their distance function uses only the diagonal elements of the samle covariance matrix. The good erformance of their rocedures aear to suggest that this indeed might be the case. To ascertain these assumtions on the covariance matrix, the likelihood ratio tests cannot be used as the samle size N = n + could be smaller than the dimension. Although, the locally best invariant LBI test roosed by John 97 and considered by Suguira 97, and Nagao 973 for the sherecity hyothesis and the LBI test given by Nagao 973 for testing the hyothesis that the covariance matrix Σ is an identity matrix can be comuted for all samle sizes, there aears to be no theoretical justification for using them as LBI tests cannot be obtained when n<. Thus, we consider a distance function between the null hyothesis and the alternative hyothesis, and roose tests based on consistent estimators of these arametric functions of the covariance matrix Σ for testing the hyothesis of sherecity of the covarinace matrix, and for testing the hyothesis that the covarinace matrix is an identity matrix. In addition, under the same set of conditions, we rovide tests for testing the hyothesis that the covarinace matrix is a diagonal matrix. Asymtotic distributions of these test statistics are given under the hyothesis as well as under the alternative hyothesis. Our focus is however for the case when n = O δ, 0 <δ. Thus, it includes the case when n/ 0. The Received October 6, 004. Revised January 4, 005. Acceted March 4, 005. *Deartment of Statistics, University of Toronto, 00 St George Street, Toronto, Canada M5S 3G3. srivasta@utstat.toronto.edu

2 5 MUNI S. SRIVASTAVA organization of the aer is as follows. In Section, we resent notations and some reliminary results. Sections 3 to 5 develo tests for testing that the covariance matrix Σ = σ I,Σ=I, and Σ = diagσ,...,σ resectively. The roofs of theorems and lemmas stated in Sections to 4 are given in Section 6.. Notations and reliminaries Let x,...,x N be indeendently and identically distributed iid as a - dimensional random vector x which is distributed as multivariate normal with mean vector µ and covariance matrix Σ, denoted, x N µ, Σ. We shall assume that Σ > 0, that is, it is a ositive definite matrix. Let x and S denote the samle mean vector and the samle covariance matrix resectively, defined as. and. x = N N α= S n V = n x α, N = n +, N α= x α xx α x resectively. Let a i = tr Σ i /, i =,...,8. We make the following assumtions: A:As,a i a 0 i, 0 <a 0 i <,,...,8. B : n = O δ, 0 <δ. In the next lemma, we give unbiased and consistent estimators of a and a. Lemma.. Under the assumtion A, and as n, an unbiased and consistent estimators of a and a are resectively given by.3 and.4 â = tr S/, â = n n + n tr S n tr S ]. For roof, see Section 6. Theorem.. Let â and â be as defined in.3 and.4 resectively. Then under the assumtion A, asymtotically â â N a a ], n a 4a 3 4a 3 8a 4 +4/na.

3 SOME TESTS CONCERNING COVARIANCE MATRIX 53 The roof is given in Section 6. Remark.. From the definition of â and â, it follows that tr S =â + tr S n =â + nâ. Thus, unless /n goes to zero as n and, tr S / is not a consistent estimator of a. That is, if n = O, tr S / is not a consistent estimator of tr Σ /, while â is always a consistent estimator of a irresective of how n, rovided the assumtion A is satisfied. Corollary.. Let n and such that /n c. Then, asymtotically tr S tr S N where = a a + ca ],n c, a 4ca a + a 3 4ca a + a 3 4a 4 + ca +4ca a 3 +c a a. The roof is given in Section 6. Let A =a,...,a q : q. Then veca isaq vector defined by veca =a,...,a q. Now, we consider some known asymtotic results. Lemma.. ns = ns ij W Σ,n, where Σ=σ ij. Then, if θ = vecσ and B =b ij, where the elements of Ω are given by vecb nvecs θ is AN 0, Ω, Eb ij b kl =σ ik σ jl + σ il σ jk, see Hsu 949, Bilodeau and Brenner 999, or Srivastava and Khatri 979, Problem 3.3,. 03. Lemma.3. Let g =g,...,g k, where g i s are differentiable functions of vec S : at θ = vec Σ. Then ngvecs gθ Nk 0,DgθΩDgθ, where g Dgθ = vecs vec S=θ : k. For roof, see Bilodeau and Brenner 999,. 79.

4 54 MUNI S. SRIVASTAVA 3. A test for the sherecity In this section, we consider the roblem of testing the hyothesis H :Σ=σ I vs A :Σ σ I, when a samle of size N = n +, x,...,x N is drawn from N µ, Σ. When n>, the most aroriate commonly used test statistic is the likelihood ratio test which has been shown by Carter and Srivastava 977 to have a monotone ower function. However, when n<the likelihood ratio test is not available. Even the cometitive locally best invariant test which can be calculated for n<, cannot be justified on theoretical grounds, although Ledoit and Wolf 00 have roosed this LBI test and have given its asymtotic distribution when /n c, a constant, but the non-null distribution is not available. In this aer, we consider a test based on a consistent estimator of a arametric function of Σ which seerates the null hyothesis from the alternative hyothesis which we discuss next. The testing roblem remains invariant under the transformation x Gx, where G belongs to the grou of orthogonal marices. The roblem also remains invariant under the scalar transformation x cx. Thus, we may assume without any loss of generality that Σ = diagλ,...,λ, a diagonal matrix. From the Cauchy-Schwarz inequality, it follows that λ i λ i with equality holding if and only if λ i λ for some constant λ. Thus γ λ i / λ i/, and is equal to one if and only if λ i = λ for some constant λ. Thus, γ is equal to one if and only if λ i = λ for some constant λ. Hence, we may consider testing the hyothesis H : γ =0 vs A : γ > 0. A test for the above hyothesis H vs A can be based on a consistent estimator of γ. From Lemma., it follows that under the assumtions A and B, a consistent estimator of γ is given by n ˆγ = tr S ] n n + n tr S /tr S/ = â â. Thus, a test for the sherecity can be based on the statistic T = ˆγ.

5 SOME TESTS CONCERNING COVARIANCE MATRIX 55 The following theorem gives the asymtotic non-null distribution of the statistic T. where Theorem 3.. Under the assumtions A and B, asymtotically n T γ + N0,τ τ = na 4a a a a 3 + a 3 a 6 + a a 4. Corollary 3.. Under the hyothesis that γ =,and under the assumtions A and B, asymototically n T N0,. Remark 3.. The reader is reminded that it is a one-sided test for testing the hyothesis that γ =vsγ >. 4. A test for the covariance matrix to be an identity matrix Let x,...,x N be iid N µ, Σ. In this section, we consider the roblem of testing the hyothesis H :Σ=I against the alternative A :Σ I. The usual likelihood ratio test for n>which has monotone ower function as shown by Nagao 967 does not exist when n<. Even, the locally best invariant test roosed by Nagao 973 cannot be justified theoretically. However, Ledoit and Wolf 00 have orosed a modified version of Nagao s test and have given its asymtotic null distribution when /n c>0, where c is a constant but the non-null distribution is lacking. In this aer, we consider a distance function between the hyothesis and the alternative hyothesis and roose a test based on a consistent estimator of this arametric function, which we discuss next. Since the roblem remains invariant under the grou of orthogonal transformations, we may assume without any loss of generality that Σ = diagλ,...,λ, a diagonal marix and test the hyothesis that λ i =, i =,..., against the alternative that λ i, for at least one i, i =,...,. But λ i = for all i if and only if Σ λ i =0. That is, λ i λ i + ] = tr Σ trσ+ ] = a a +=0.

6 56 MUNI S. SRIVASTAVA Thus, a test for the hyothesis H :Σ=I vs A :Σ I, can be based on a consistent estimator of 4. γ = a a. From Lemma., it follows that under the assumtions A and B, an unbiased and consistent estimate of γ is given by n ˆγ = n n + =â â, and our test will be based on the test statistic tr S ] n tr S tr S] 4.4 T =ˆγ +. where Theorem 4.. Under the assumtions A and B, asymtotically n T γ N0,τ, τ = n a a 3 + a 4 +a. The roof is given in Section 6. Corollary 4.. Under the hyothesis H : Σ=I, and the assumtions A and B, the asymtotic null distribution of T is given by n T N0,. Remark 4.. This is a one-sided test for testing H : γ +=0vsA : γ +> 0. Remark 4.. For testing the hyothesis that Σ = Σ 0, we consider the observations y i =Σ / 0 x i, and test the hyothesis that the covariance matrix of y i is an identity matrix. Here Σ = Σ / Σ /. 5. A test for the covariance to be diagonal Let x,...,x N be iid N µ, Σ. We wish to test the hyothesis H : σ ij =0, i j, i, j =,...,, against the alternative A : σ ij 0, for at least one air i, j, i j, where Σ = σ ij. Without loss of generality, we may consider the hyothesis H : ρ ij =0vsA : ρ ij 0 for at least one air i, j, i j,

7 SOME TESTS CONCERNING COVARIANCE MATRIX 57 i, j =,...,where ρ ij = σ ij /σ ii σ jj /, i j. Let r ij be the samle correlation coefficient defined by 5. r ij = s ij sii s jj = v ij vii v jj, i j, i, j =,...,, where S =s ij and ns = V =v ij. Define 5. and q =, r =r,r 3,...,r,, ρ =ρ,ρ 3,...,ρ,. Then it follows from the results of Hsu 949, that as n, nr ρ Nq 0, Ω, where the diagonal elements of Ω are given by ρ, ρ 3,..., ρ,. The off diagonal elements are given by 5.3 covr ij,r kl =ρ ij ρ kl + ρ kl ρ il ρ lj ρ ij ρ kj + ρ il ρ kl ρ ki ρ ij ρ il + ρ kj ρ kl + ρ kiρ lj ρ ij + ρ il + ρ kj + ρ kl. It may be noted that when ρ ij =0,Ω=I, and thus nγ N q 0,Ias n. Since ρ ij =0 if and only if ρ ij = 0, a test for the hyothesis H : ρ ij = 0 against the alternative A : ρ ij 0 for at least one air of i, j, i j, can be based on the test statistic 5.4 T3 = n r ij q, q =. q Since, under the hyothesis H : ρ ij =0,nrij are asymtotically indeendently distributed with mean and variance, it follows from the central limit theorem that as, q n r ij q N0,. Thus, asymtotically n r ij q q N0,. Thus, we get the following theorem

8 58 MUNI S. SRIVASTAVA Theorem 5.. go to infinity Under the hyothesis H : ρ ij =0,asymtotically as n and T 3 N0,. Under the alternative A : ρ ij = n / c ij, n / <c ij <n /,Ω=I + On /. Hence, we get the following corollary. Corollary 5.. Under the alternative ρ ij = n / c ij, asymtotically ] q / q +c T3 N c c c q +c c,. The asymtotic normality of T3 given in Theorem 5. is rather slow. Thus, as an alternative, Chen and Mudholkar 990 considered the Fisher s z-transformation z ij, defined by 5.5 z ij = log e +r ij r ij and roosed a test based on the test statistic 5.6 T3 =n zij. They aroximated its distribution by a Xv + b where v, a and b are obtained by equating the first three moments of T3 with that of axv + b where Xv is a chi-square random variable with v degrees of freedom. However, neither of the two tests have been desgined for large. For examle, the values of a, v, and b deends only on n. Thus, we consider another arametric measure of the deviation of the hyothesis from the alternative and roose a test based on its consistent estimate. Let Then a 0 = σ n ii, â 0 = s ii n + a 40 = σ ii, 4 â 40 = s 4 ii. a = tr Σ = σii + σij i j = a 0 + σ ij. i j

9 SOME TESTS CONCERNING COVARIANCE MATRIX 59 We consider the arametric function 5.9 γ 3 =a /a 0. Clearly γ 3 = if and only if σ ij = 0. And if σ ij 0,γ 3. Thus, we can base our test on a consistent estimator of γ 3 given by â 5.0 ˆγ 3 = where â is defined in.4, and â 0 is given in 5.7. We will be testing the hyothesis H : γ 3 = 0 against the alternative A : γ 3 0. Thus, it will also be a one-sided test. We note that tr S ] n tr S 5. ˆγ 3 = = = n n n n n n s ii i j s ij + â 0 s ii n i j s ij n s iis jj s ii s ii s ii n i j s iis jj ] + n n n i j s ij n s iis jj = n +, s ii which is a statistic based on the samle covariances and not on samle correlations. This aears to be a reasonable rocedure as when n<, the samle covariance matrix S is singular, and in this case the diagonal elements s ii may be very small and may lead to larger values of r ij. From the asymtotic theory given in Theorem 6., it follows that ˆγ 3 N γ 3, a 0 ] 4 n a a 4. Thus, we roose a test based on the test statistic n ˆγ 3 5. T 3 = ] /. â40 In the next theorem, we give an asymtotic distribution of T 3. Theorem 5.. Under the assumtions A and B, the asymtotic distribution of T 3 is given by T 3 Nδ, τ3 ], â 0

10 60 MUNI S. SRIVASTAVA where and δ = n γ 3 ] / a40 a 0 τ 3 =a a 4 /a 0 a 40. Since under the hyothesis H, δ =0,and τ 3 =,we get the following corollary. Corollary 5.. Under the hyothesis H : σ ij =0,and under the assumtions A and B, T 3 is asymtotically distributed as N0,. 6. Proofs In this section, we rove lemmas and theorems stated in Sections to 4. But before we begin these roofs, we state some results on the moments of a chi-square random variable with n degrees of freedom, and other reliminary results. 6.. Preliminary results We begin with the following lemma. Lemma 6.. Then Let v be a chi-square random variable with n degree of freedom. Ev r =nn + n +r, r =,,... Varv =n, Varv =8nn + n +3, Ev n 3 =8n, Ev n 4 =nn +4, Ev nn + ] 4 =3nn + 7n 4 + On 3 ]. Next, we obtain exressions for tr S, trs and tr S in terms of chi-square random variables. Let V = ns = YY W Σ,n, where Y =y,...,y n and y i are iid N 0, Σ. Let Γ be an orthogonal matrix such that ΓΣΓ =, where = diagλ,...,λ and λ i are the eigenvalues of Σ. Then, if U =u,...,u n, where u i are iid N 0,I, Y =Σ / U, and Σ / Σ / =Σ, tr S = n tr U ΣU = n tr W W = λ i w n iw i,

11 SOME TESTS CONCERNING COVARIANCE MATRIX 6 where U Γ = W =w,...,w, and w i are iid N n 0,I. Thus, if v ii = w i w i, v ii are iid chi-square random variables with n degrees of freedom. Hence, and 6.3 nâ = ntr S = λ i v ii, n trs = λ i vii + λ i λ j v ii v jj, n tr S = trw W W W ] =tr λ i w i w i λ i w i w i =tr λ i w i w iw i w i + λ i λ j w i w iw j w j = λ i w iw i + λ i λ j w iw j = λ i vii + λ i λ j vij, where 6.4 v ij = w iw j, i j, v ii = w iw i. Thus, since n /n n +, â tr S ] 6.5 n tr S = n λ i vii + λ i λ j vij λ i vii + λ i λ j v ii v jj n = n b n λ i vii + λ i λ j vij n n v iiv jj where n = n 3 = q + q, ] λ i vii + n λ i λ j z ij 6.6 z ij = v ij n v iiv jj =w iw j n w iw i w jw j.

12 6 MUNI S. SRIVASTAVA From now on, we shall consider only the asymtotic version of â, without the sign Lemma 6.. For z ij defined above, we have 6.7 Ez ij =0, Ez ij z ik =0, for all distinct i, j, k, Varz ij =n + n. Proof. Since w i are iid N n 0,I n, it follows that Ez ij =E w iw j w jw i ] n w iw i w jw j = Ew iw i n n n =0, as Ew j w j =I n and Ew i w i=n. Similarly, for all distinct i, j, k, Ez ij z ik ]=E w j w i ] n w i w i w j w j w k w i ] n w i w i w k w k = E w j w i w k w i n w k w i w i w i w j w j n w j w i w i w i w k w k + ] n w i w i w j w j w k w k. Thus, and, Ew j w i w k w i ]=Ew j w i w i w j w i w k w k w i ] = E{w i w j w j w i }{w i w k w k w i }] = Ew i w i w i w i ] = nn +, Ew i w i w j w j w i w k w k w i ] = Ew i w i w j w j w i w i ] = n n +, Ew i w i w k w k w i w j w j w i ] = new i w i w k w k ] = n n +, Ew i w i w j w j w k w k ] = n 3 n +. Hence, for all distinct i, j, k Ez ij z ik ]=0. To calculate the variance of z ij, we note that and z ij = v 4 ij n v ijv ii v jj + n v iiv jj, i j,

13 SOME TESTS CONCERNING COVARIANCE MATRIX 63 Ev 4 ij =Ew i w j w j w i = Ew i A j w i, where A j = w j w j. Let G be an orthogonal matrix such that GA j G = diagw j w j, 0...,0. Since given A j, l i = Gw i N n 0,I, it follows that l i is indeendently distributed of A j, and Evij 4 =Eziw 4 j w j ] =3nn +, where l i =l i,...,l ni = Gw i N n 0,I. Thus, l i N0, and is indeendently distributed of w j w j. Similarly, Evijv ii v jj =Ew i w j w j w i w i w i w j w j = Ew i A j w i w i w i tra j, A j = w j w j n ] = E liw j w j w j w j = Ew j w j E = nn + n + = nn +. lki k= n li lki k= Finally, Hence, Ev iiv jj =n n +. Varz ij =3nn + n + +n + =3nn + n + =n + n. Lemma 6.3. Let v ii be iid as χ n, 6.8 q = n n 3 λ i v ii, and q = n λ i λ j z ij. Then, Covq,q =0 Varq = 8n + n + 3n /n 5 ]a 4 Varq = 8n + n /n 4 ] λ i λ j/ 4/n a a 4 /].

14 64 MUNI S. SRIVASTAVA Proof. We note that Since Eq = 0, we get n 5 n 5 Covq,q = Eq q n n = E λ i vii] λ j λ k z jk = λ E j<k v λ j λ k z jk j<k v λ j λ k z jk j<k v j<k + λ E + + λ E λ j λ k z jk. Eviiz jk,j k] { = E w i w i w j w k }] n w j w j w k w k = E w i w i w j w k w k w j ] n w i w i w j w j w k w k = E w i w i 3 ] n w i w i 3 w k w k =0, i = j k = E w i w i w j w j ] n w i w i w j w j w k w k =0, i j k = E w i w i w i w j w j w i ] n w i w i 3 w j w j =0, j k = i. The variances of q and q can easily be obtained. Note that, λ i λ j = 6. λ i λ = i 4 λ 4 i = a a 4 /, λ 4 i which is bonded under the assumtion A. Hence, we get the following corollary.

15 SOME TESTS CONCERNING COVARIANCE MATRIX 65 Corollary 6.. As n, q 0 in robability under the assumtion A. Thus, also under the assumtions A and B, q 0 in robability as. Then Corollary 6.. Let â = n λ i v ii = Covq, â =0. tr S. Lemma 6.4. Let ns W Σ,n, and â = tr S ] n tr S = q + q. Then, from 6.0, and 6. and the fact that Covq,q =0, Varâ =Varq +Varq 8n + n + 3n 4n + n = n 5 a 4 + n 4 a a 4 ] 8a 4 n + 4 n a a 4 ] 4 a n + n ] a 4. Lemma 6.5. Let v ii be iid as χ n, q =n n 3 λ i v ii, and â = n λ iv ii. Then Covq, â =4n n +a 3 /n 3 4a 3 /n as n. Proof. We have n 4 n Covq, â ] = E λ i vii λ i v ii n n + a a = E λ 3 i vii 3 + λ i viiλ j v jj n n + a a i j = nn + n +4a 3 + n n + λ i λ j n n + a a i j =nn + n +4a 3 n n +a 3 ] =4nn +a 3 ].

16 66 MUNI S. SRIVASTAVA Next, we obtain the variance of ˆγ. We have ˆγ tr S ] 6.3 n tr S trs = n λ n 3 i vii + n W ij λi v ii n where 6.4 Now 6.5 n λ i v ii n λi v ii + n = n λ i v ii nv ii λ i ]+ W ij = λ i λ j z ij. W ij n W ij, Varλ i vii nv ii λ i ] = λ 4 i Varvii 4nλ 3 i Covvii,v ii +4n λ i Varv ii =8nn + n +3λ 4 i +8n 3 λ i 4nλ 3 i Ev ii nvii] =8nn + n +3λ 4 i +8n 3 λ i 4nλ 3 i Evii 3 nvii] Hence, =8nn + n +3λ 4 i +8n 3 λ i 4nλ 3 i nn + n +4 n n + ] =8nn + n +3λ 4 i +8n 3 λ i 6n n +λ 3 i. Var tr S ] n tr S trs ] Var λ n i vii nv ii λ i +Var n W ij + Cov n λ i vii nv ii λ i, n W ij = 8nn + n +3 λ 4 i +8n3 λ i 6n n + λ 3 i n 4 4n + n + n 4 λ i λ j +0 i j = 8nn + n +3a 4 +8n 3 a 6n n +a 3 n 4 4n + n a ] a 4 + n 4 8a 4 +8a 6a a n n ] a 4.

17 SOME TESTS CONCERNING COVARIANCE MATRIX 67 Hence, we get the following lemma Lemma 6.6. For ˆγ defined above, Varˆγ 8a 4 6a 3 +8a n + 4 n a a 4 ]. Corollary 6.3. When λ i =, Varˆγ 4, for large n and. n Let 6.6 and 6.7 u i = λ iv ii n, u i = λ i v ii nn + ], n nn + n +3 ε n =n +/n + 3] /. Then 6.8 Eu i =0, Eu i =0 Varu i =λ i, Varu i =8λ 4 i, Covu i,u i =4ε n λ 3 i. Thus u i = u i u i are indeendently distributed random vectors, i =,...,, with mean vectors as zero vectors and the covariance matrices M in given by 6.9 Now, as M in = λ i 4ε n λ 3 i 4ε n λ 3 i 8λ 4 i, i =,...,. 6.0 where 6. M n M n M n a 4ε n a 3 = Mn 0 0, for any n, 4ε n a 3 8a 4 a Mn 0 0 = 0 4ε n a 0 3 a 0 4ε n a 0 3 8a 0 0 4a a 0 M 0 as n. 3 8a0 4

18 68 MUNI S. SRIVASTAVA Also, if F i is the distribution function of u i, then u udf i ε u u df i u u>ε = ε Eu i + u i ε from c r -inequality, see Rao 973,. 49. Now Eu 4 i + u 4 i, Similarly Eu 4 i = λ 4 Ev i n 4 i n nn +4 a 4 = n 0 as. Eu 4 i = λ 8 Evi nn + 4 i n n + n +3 = O 0 as. Then from the multivariate central limit theorem of Liaunov tye given in Rao 973,. 47, Problem 4.7, it follows that as, and for any n, u i = λ iv ii n n λ i v ii nn+ N 0,Mn. 0 n+n+3 Thus, it follows that as and then n, u i N 0,M 0. On the otherhand, as n, we get from the multivariate central limit theorem that u i N O, M i, i =,..., for any, where M i is the limit of M in given by λ M i = i 4λ 3 i 6. 4λ 3 i 8λ 4. i Let 6.3 M = M a 4a M =, 4a 3 8a 4

19 SOME TESTS CONCERNING COVARIANCE MATRIX 69 which goes to M 0 as. Since u i are asymtotically indeendently distributed, it follows from the argument given above that as n, and then u i N 0,M 0. Without any loss of generality, we may relace M 0 by M. Noting that and â =n λ i v ii, q = n n 3 λ i vii n λ i vii we get the following theorem Theorem 6.. As n and, in any manner, ] â a N, n M. q a Corollary 6.4. For any finite n, and as, ] â a N, n M n. q a Theorem 6.. Let z ij be as defined in 6.6, and q = n λ i λ j z ij. Then under the assumtions A and B, and as n and, q N0, 4n a a 4 ]. Proof. Note that z ij =w i w j n w i w i w j w j and = v ij n v iiv jj, n z ij =n v ij n v ii v jj n.

20 70 MUNI S. SRIVASTAVA The second term goes to zero in robability and Covn v ij n,n v ik n] 0asn for all distinct i, j, k. Since n / v ij N0, as n, it follows that n vij is a chi-square random variable with one degree of freedom and are asymtotically indeendently distributed for all distinct i and j. Now, when, we aly Liaunov tye central limit theorem to obtain the asyntotic normality of q. Because of the normality assumtion, the same result is obtained if we interchange the order of limit. Next, we note that Covâ,q = 0 and Covq,q = 0. From the asymtotic normality of â,q and the fact that the covariance between â,q and q is a zero vector, it follows that â,q,q are jointly asymtotically normally distributed as stated in the following theorem. Theorem 6.3. Under the assumtions A and B, asymtotically â a q N a, n M 0 0 4n a 0 a 4. q 6.. Proof of Lemma. From 6.5, and Lemma 6. Eâ = E tr S n tr S ] = n ] n 3 E λ i vi + n E = n n 3 nn + λ i + n = n n + n tr Σ. λ i λ j z ij λ i λ j Ez ij Hence, 6.4 n tr S ] n n + n tr S is an unbiased estimator of tr Σ /. From Lemma 6.3, q goes to zero in robability as n. Thus it follows from Lemma 6.4 and equation 6.5 that 6. is under assumtion A, an unbiased and consistent estimator of a as n, and â = tr S ] n tr S is a consistent estimator of a as n goes to infinity and the assumtion A holds.

21 SOME TESTS CONCERNING COVARIANCE MATRIX Proof of Theorem. and Corollary. Since â = q + q and â = λ iv i /n] = trs/, it follows from Theorem 6.3 that ] â a N, n a 4a 3 â a 4a 3 8a n a To rove Corollary., we note that c = /n, g â, â tr S =â, and g â, â tr S =â + câ. Hence, from Lemma.3, the covarinace of g,g is given by g g n â â a 4a g g 3 â â g g â â 4a 3 8a 4 +4ca g g â â 0 =n a 4a 3 a c a c 4a 3 8a 4 +4ca 0 = n c a 4ca a + a 3 4ca a + a 3 4c a a +4ca a 3 + ca +a Proof of Theorem 3. We note that Hence, T â Thus, asymtotically, where τ a =, a = a 3 T = â â. = â â 3, and a n 4a 3 n T Nγ,τ 4a 3 n 8a 4 n + 4a n 4a na 3 + 4a 3 na, 8a a 3 na 3 = 8a3 na 6 6a a 3 na 5 + a 4 T â a /a 3 a = â. + 8a4 a n + 4a n 8a4 n + 4a n = 8a3 na 6 6a a 3 na 5 + 8a 4 na 4 + 4a n a 4. Thus, when λ i = λ, τ =4/n. ] a /a 3 a

22 7 MUNI S. SRIVASTAVA 6.5. Proof of Theorem 4. Since ˆγ =â â it follows from Theorem 6.3 that asymtotically T Nγ +,τ where Thus, when λ i =, τ = 8a 4 6a 3 +8a n τ = 4 n. + 4a n. Acknowledgements I wish to thank the two referees for their careful reading of the article and for their suggestions. This research was suorted by the Natural Sciences and Engineering Research Council of Canada. References Bilodeau, M. and Brenner, D Theory of Multivariate Statistics, Sringer-Verlag, New York. Carter, E. M. and Srivastava, M. S Monotonicity of the ower functions of the modified Likelihood ratio criteria for the homogeneity of variances and of the shericity test, J. Multiv. Anal., 7, Chen, S. and Mudholkar, G. S Null distribution of the sum of squared z-transformations in testing comlete indeendence, Ann. Inst. Statist. Math., 4, Dudoit, S., Fridlyand, J. and Seed, T. P. 00. Comarison of discrimination methods for the classfication of tremors using gene exression data, J. Amer. Statist. Assoc., 97, Hsu, P. L The limiting distribution of functions of samle means and alications to testing hyothesis, Proceedings of the First Berkeley Symosium of Mathematical Statistics and Probability ed. J. Neyman, Univ. of California Press, Berkeley and Los Angeles, John, S. 97. Some otimal multivariate tests, Biometrika, 58, 3 7. Ledoit, O. and Wolf, M. 00. Some hyothesis tests for the covariance matrix when the dimension is large coared to the samle size, Ann. Statist., 30, Nagao, H Monotonicity of the modified likelihood ratio test for a covariance matrix, J. Sci. Hiroshima Univ., A3, Nagao, H On some test criteria for covariance matrix, Ann. Math. Statist.,, Rao, C. R Linear Statistical Inference and Its Alications, nd ed., Wiley, New York. Srivastava, M. S. and Khatri, C. G An Introduction to Multivariate Statistics, North- Holland, New York. Sugiura, N. 97. Locally best invariant test for sherecity and the limiting distributions, Ann. Math. Statist., 43, 3 36.