Journal of Multivariate Analysis 00 (009) 305 3 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva Sphericity test in a GMANOVA MANOVA model with normal error Peng Bai Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650, PR China a r t i c l e i n f o a b s t r a c t Article history: Received 4 May 008 Available online 5 March 009 AMS 000 subject classifications: 6E5 6E0 6H0 6H5 Keywords: GMANOVA MANOVA model Sphericity test Null distribution Meijer s G p,0 p,p function Asymptotic distribution Bartlett type correction For the GMANOVA MANOVA model with normal error: Y = XB Z + B Z + E, E N q n (0, I n Σ), we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: Σ = λi q (λ is unknown). It is shown that, as a function of the likelihood ratio statistic Λ, the null distribution of Λ /n can be expressed by Meijer s G q,0 q,q function, and the asymptotic null distribution of log Λ is χ q(q+)/ (as n ). In addition, the Bartlett type correction ρ log Λ for log Λ is indicated to be asymptotically distributed as χ q(q+)/ with order n for an appropriate Bartlett adjustment factor ρ under null hypothesis. 009 Elsevier Inc. All rights reserved.. Introduction The model considered here is a GMANOVA MANOVA model which can be defined as Y = XB Z + B Z + E, where Y is a q n observable random response matrix, X is a q p known constant matrix, Z and Z are the n m and n s known design matrices, respectively, B and B are the p m and q s unknown regression coefficient matrices, respectively, E is a q n unobservable random error matrix, and A denotes the transpose of matrix A. The model () was first proposed by Chinchilli and Elswick [], and was extensively applied to various fields including biology, medicine and economics. The error matrix E is often assumed to be normal: E N q n (0, I n Σ), i.e. E,..., E n i.i.d N q (0, Σ)(E ˆ=(E,..., E n )), where Σ(> 0) is a q q unknown covariance matrix. Under the assumption (), a variety of investigations have been made to handle the statistical inferences with respect to the parameter matrices B, B and Σ, a good summary for the related results can be found in Kollo and von Rosen [], and the excessive published papers will not be listed here for being irrelative to our subject. The available materials clearly show that most of the published works relating to the model () and () focused their attention on the statistical inferences for B and B, and few took Σ into account. In this paper, we study an inference with respect to Σ, which is referred to as sphericity hypothesis and can be described as H : Σ = λi q, λ(> 0) is unknown. () () (3) E-mail address: baipeng68@hotmail.com. 0047-59X/$ see front matter 009 Elsevier Inc. All rights reserved. doi:0.06/j.jmva.009.03.003
306 P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 To the best of our knowledge, the likelihood ratio test for the above hypothesis in the model () under the assumption () has not been done before. The remainder of this article is arranged as follows: Section gives the likelihood ratio statistic Λ for sphericity hypothesis (3). In Section 3, the exact null density function of Λ /n is expressed by Meijer s G q,0 q,q function, the asymptotic null distribution of log Λ is shown to be χ q(q+)/ (as n ), and ρ log Λ is indicated to be asymptotically distributed as χ q(q+)/ with order n for an appropriate Bartlett adjustment factor ρ for log Λ under null hypothesis.. Likelihood ratio statistic In order to obtain the likelihood ratio test statistic for sphericity hypothesis (3), we need the following results. We follow the symbols and notations in Muirhead [3] without specification. Lemma (Bai [4]). For the GMANOVA MANOVA model () with normal error (), the maximum likelihood estimates of B, B and Σ are given by (with probability one) ˆB = (X S X) X S YQ Z Z (Z Q Z Z ), ˆB = (Y X ˆB Z )Z (Z Z ), ˆΣ = n (Y X ˆB Z )Q Z (Y X ˆB Z ), respectively, where S = YQ Z Y, Z ˆ= (Z, Z ), P A ˆ= A(A A) A, Q A ˆ= I p P A (A is a p q matrix) and A denotes an arbitrary g-inverse of A such that AA A = A. Remark. The sufficient and necessary conditions for the random matrix S in Lemma being positive definite with probability one are n rk(z) + q (Okamato [5]), where rk(a) denotes the rank of matrix A. In addition, although the expressions of both ˆB and ˆB contain the g-inverses, we have ˆΣ = n {S + [I q X(X S X) X S ](S S)[I q X(X S X) X S ] }, (4) which and R(X ) = R(X S ) show that ˆΣ is unique, where S = YQ Z Y and R(A) denotes the linear subspace spanned by the columns of matrix A. Lemma. Let K(B, B, Σ Y) denote the likelihood function of (B, B, Σ) based on Y in the model () and (), i.e. { K(B, B, Σ; Y) = (π) qn/ Σ n/ etr (Y XB Z B Z ) Σ (Y XB Z B Z ) }, B R p m, B R q s, Σ > 0, (5) then sup K(B, B, λi q ; Y) = (πeˆλ) qn/, (6) B R p m,b R q s,λ>0 where ˆλ = qn tr(p X S + Q X S ). Proof. It follows from (5) that L(B, B, λ; Y) ˆ= log K(B, B, λi q ; Y) which implies that = qn log(πλ) λ tr{ (Y XB Z B Z )(Y XB Z B Z ) }, L(B, B ; Y) ˆ= sup L(B, B, λ; Y) λ>0 B R p m, B R q s, λ > 0, (7) = qn log{πe λ(b, B )}, B R p m, B R q s, (8)
P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 307 where λ(b, B ) = tr{(y XB qn Z B Z )(Y XB Z B Z ) } and tr(a) denotes the trace of matrix A. Note that (Y XB Z B Z )(Y XB Z B Z ) = (Y XB Z )Q Z (Y XB Z ) + (B B (B ))Z Z (B B (B )), B R q s, B R p m, where B (B ) = (Y XB Z )Z (Z Z ), hence tr{(y XB Z B Z )(Y XB Z B Z ) } tr{(y XB Z )Q Z (Y XB Z ) }, B R p m, (9) where the equality holds if B = B (B ), B R p m. Again note that (Y XB Z )Q Z (Y XB Z ) = [Y X ˆB0 Z X(B ˆB0 )Z ]Q Z [Y X ˆB0 Z X(B ˆB0 )Z ] = [Q X YP QZ Z X(B ˆB0 )Z Q Z ][Q X YP QZ Z X(B ˆB0 )Z Q Z ] + S, B R p m, where ˆB0 = (X X) X YQ Z Z (Z Q Z Z ), thus tr{(y XB Z )Q Z (Y XB Z ) } = tr(p X S + Q X S ) + tr{x(b ˆB0 )Z Q Z Z (B ˆB0 )X } tr(p X S + Q X S ), (0) where the equality holds if B = ˆB0. From the definition of λ(b, B ), (9) and (0), we have λ(b, B ) ˆλ ˆ= qn tr(p X S + Q X S ), B R p m, B R q s, where the equality holds if B = B (B ), B = ˆB0. This and (8) show that sup L(B, B, λ; Y) = sup L(B, B ; Y) B R p m,b R q s,λ>0 B R p m,b R q s qn log(πeˆλ), where the equality holds if B = B (B ), B = ˆB0. Therefore, from (7), we obtain (6). From Lemmas and, we immediately have Corollary. For the model () and (), the likelihood ratio statistic for testing the sphericity (3) is n/ ˆΣ Λ =, ˆλ q () where ˆΣ and ˆλ are given by Remark and Lemma, respectively. Proof. It follows from Lemma and (5) that sup K(B, B, Σ; Y) = (πe) qn/ ˆΣ n/, B R p m,b R q s,σ>0 which and (6) mean that the likelihood ratio statistic for testing the sphericity (3) is given by (). 3. Null distribution In this section, we will establish the exact null density function of Λ /n and the asymptotic null distributions of log Λ. The following theorem plays the key role for deriving the results mentioned above. Theorem. The null distribution of likelihood ratio statistic Λ is determined by Λ /n d = q q U V [tr(u) + tr(v) + W] q, () where X = d Y denotes that the random variables X and Y have the same distribution, U, V, W are mutually independent and U W rk(x) (n, I rk(x) ), V W q rk(x) (n, I q rk(x) ), (3) W χrk(x)(q rk(x)),
308 P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 where n = n rk(z) q + rk(x) and n = n rk(z ). Proof. Let the singular value decomposition of X be 0 X = P Q, 0 0 (4) where P and Q are the q q and p p orthogonal matrices, respectively, is a rk(x) rk(x) nonsingular diagonal matrix, then Irk(X) 0 P X = P P 0 0, Q 0 0 X = P P. (5) 0 I q rk(x) Make the transformation T T T ˆ= = P SP, T T (6) where T is a rk(x) rk(x) random matrix, then ( S = PT P = P T T T T T T T T ) P, (7) where T = T T T T, T = T T T T, hence from (4) and (7), we have ( X S T X = Q ) 0 Q, 0 0 which means that (X S X) = T Q C Q, (8) C C where C, C and C are arbitrary. Substitute (4), (6) (8) into (4) to yield ˆΣ = ] [T n P 0 T + T P 0 0 ( S S)P 0 I q rk(x) T T P, (9) I q rk(x) where S = EQ Z E, S = EQ Z E. Furthermore, make the transformation ) ( T T T ˆ= = P ( S S)P, T T (0) where T is a rk(x) rk(x) random matrix, then from (6) and (9), we obtain ˆΣ = ( n P T + T T T T T T (I q rk(x) + T T ) ) (I q rk(x) + T T )T P, T + T which shows that (Theorem A5.3 in [3]) ˆΣ = n q T T + T. () In addition, it follows from (5), (6) and (0) that ˆλ = qn tr{s + Q X( S S)} = qn [tr(t ) + tr(t T T ) + tr(t + T )]. () When the sphericity hypothesis (3) holds, from (), we have E N q n (0, λi n I q ), which implies that { S Wq (n rk(z), λi q ), S S W q (rk(z) rk(z ), λi q ). (3) (4)
P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 309 Note that Q Z (Q Z Q Z ) = 0, hence from (3) and Theorem 0.4 in Schott [6], S = EQ Z E and S S = E(Q Z Q Z )E are mutually independent. Therefore, from (6), (0) and (4), we know that T and T are mutually independent and { T Wq (n rk(z), λi q ), T W q (rk(z) rk(z ), λi q ), which and Theorem 3..0 in Muirhead [3] indicate that T W rk(x) (n, λi rk(x) ), T T N rk(x) (q rk(x)) (0, λt I rk(x) ), T W q rk(x) (n rk(z), λi q rk(x) ), T W q rk(x) (rk(z) rk(z ), λi q rk(x) ), (5) and T is independent of (T, T ). It follows from the independence between T and T that (T, T, T ) and T are independent, hence from the independence between T and (T, T ), we know that T, (T, T ), T are mutually independent. Note that from the second equality in (5), we have T T T T W rk(x) (q rk(x), λi rk(x) ), which means that T T T and T are independent and T T T W rk(x) (q rk(x), λi rk(x) ). Thus from the independence among T, (T, T ) and T, we know that T, T T T, T, T are mutually independent. Let U = λ T, V = λ (T + T ), (7) W = λ tr(t T T ), then U, V, W are mutually independent, and from (5) and (6), we obtain (3). Finally, it follows from (), () and (7) that λ q ˆΣ = U V, ˆλ = λ [tr(u) + tr(v) + W], n qn which and Corollary shows that () holds. In order to obtain the exact null density function of Λ /n based on Theorem, we need the following definition and lemma. Definition. If the m m nonnegative definite random matrix X has the density function ( ma Γ m (a) Σ a x a (m+)/ e tr ) Σ x (dx), x > 0, (8) where Re(a) > (m ), Σ is a m m symmetric matrix such that Re(Σ) > 0, then X is said to have an m-variate gamma distribution with parameter (a, Σ) and is denoted by X Γ m (a, Σ). When a = n, n is an integer, Γ m(a, Σ) is the Wishart distribution W m (n, Σ) (Muirhead [3]). (6) Lemma 3. If X Γ m (a, Σ), then tr(σ X) Γ (ma, ). Proof. Make the transformation Σ / XΣ / = T T, where T = (T ij ) m m is upper-triangular with positive diagonal elements, then (Theorem..9 in Muirhead [3]) m m (dx) = Σ (m+)/ m dt ij, i= T m+ i ii i j which and (8) means that the joint density function of T ij, i j m can be written as [ ] m m exp t ij dtij (π) / a (i )/ Γ [a (i )/] (t ii )a (i+)/ exp t ii dt ii, i<j i= (9)
30 P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 which shows that T ij N(0, ), i < j m, T ii Γ (a (i ), ), i =,..., m and T ij, i j m are mutually independent. Therefore, from (9) and additivity of gamma distribution [7], we have m tr(σ X) = tr(t T) = T ij Γ (ma, ). i j Theorem. Let Λ = Λ /n, when the sphericity hypothesis (3) holds, we have E( Λ z ) = q Γ qz rk(x)(n / + z) Γ q rk(x) (n / + z) Γ (n 3 /), Re(z) 0, (30) Γ rk(x) (n /) Γ q rk(x) (n /) Γ (n 3 / + qz) where n 3 = rk(x)(n rk(z)) + (q rk(x))(n rk(z )). Proof. When the sphericity hypothesis (3) holds, from Theorem, we know that { } U E( Λ z V z z ) = q qz E [tr(u) + tr(v) + W] qz = (q) Γ { qz rk(x)(n / + z) Γ q rk(x) (n / + z) E Γ rk(x) (n /) Γ q rk(x) (n /) [tr(ũ) + tr(ṽ) + W] qz }, Re(z) 0, (3) where Ũ, Ṽ, W are mutually independent and Ũ Γ rk(x) n + z, I rk(x), Ṽ Γ q rk(x) n + z, I q rk(x). (3) It follows from the additivity of gamma distribution [7], Lemma 3, (3) and (3) that tr(ũ) + tr(ṽ) + W Γ n 3 + qz,, which indicates that { } E [tr(ũ) + tr(ṽ) + W] qz Substitute the above equality into (3) to get (30). Γ = qz (n 3 /), Re(z) 0. Γ (n 3 / + qz) Theorem 3. As the function of likelihood ratio statistic, the null density function of Λ = Λ /n can be expressed as f Λ ( λ) = (π)(q )/ q (n 3 )/ π [rk(x)(rk(x) )+(q rk(x))(q rk(x) )]/4 Γ (n 3 /) G q,0 q,q Γ rk(x) (n /)Γ q rk(x) (n /) ( λ a,...,a q b,...,b q ), 0 < λ <, (33) where G p,q m,n(z a,...,a p b,...,b q ) denotes the Meijer s G-function, a i = (n i ), i rk(x), a i = (n i+rk(x) ), rk(x)+ i q, b i = q [n 3 + (i )], i q. Proof. It follows from Theorem that the Mellin transform of null density of λ is g Λ (z) = E( Λ z ) = q q(z ) Γ rk(x)(n / + z ) Γ q rk(x) (n / + z ) Γ (n 3 /), Re(z). (34) Γ rk(x) (n /) Γ q rk(x) (n /) Γ [n 3 / + q(z )] From Gauss s multiplication formula for gamma function [8], we have [ ] Γ n 3 + q(z ) = q(n 3 )/+q(z ) q [ ] Γ q (n 3 + i) + z. (35) On the other hand, (π) (q )/ i=0 rk(x) [ ] Γ rk(x) n + z = π rk(x)(rk(x) )/4 Γ (n i ) + z, (36) i=
P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 3 q rk(x) [ ] Γ q rk(x) n + z = π (q rk(x))(q rk(x) )/4 Γ (n i ) + z. (37) Substitute (35) (37) into (34) to get (π)(q )/ π [rk(x)(rk(x) )+(q rk(x))(q rk(x) )]/4 Γ (n 3 /) g Λ (z) = q (n 3 )/ Γ rk(x) (n /)Γ q rk(x) (n /) rk(x) i= Γ [(n i )/ + z] q rk(x) q i=0 i= i= Γ [(n 3 + i)/(q) + z ] Apply the definition of Meijer s G-function [9], the above equality and f Λ ( λ) = i λ z g Λ πi (z)dz, 0 < λ <, i we obtain (33). Γ [(n i )/ + z], Re(z). Remark. Davis [0] provided an effective algorithm for computing the quantile of distribution with the density expressed by Meijer s G p,0 p,p function. Furthermore, we have Theorem 4. When the sphericity hypothesis (3) holds, log Λ L χ q(q+)/, n, (38) where L denotes convergence in distribution. Proof. It follows from Theorem that the characteristic function of log Λ under hypothesis H is ϕ log Λ (t) = E(Λ it ) = q Γ iqnt rk(x)(n / int) Γ q rk(x) (n / int) Γ (n 3 /) Γ rk(x) (n /) Γ q rk(x) (n /) Γ (n 3 / iqnt), k= t (, + ), which shows that rk(x) { [ ] [ ]} log ϕ log Λ (t) = iqnt log q + log Γ (n k + ) int log Γ (n k + ) q rk(x) { [ ] [ ]} + log Γ (n k + ) int log Γ (n k + ) k= + log Γ n 3 log Γ n 3 iqnt, t (, + ). (39) Use the asymptotic formula for log(z + a) [3], ( log Γ (z + a) = z + a ) log z z + log(π) + O(z ) it is a simple matter from (39) to show that log ϕ log Λ (t) [ ] q(q + ) log( it), n, which indicates that (38) is true. In order to obtain and improve the order by approximating the null distribution of log Λ with χ q(q+)/ (as n ) based on Theorem 4, it follows from Theorem that, under the null hypothesis (3), we have E(Λ z ) = E( Λ nz/ )
3 P. Bai / Journal of Multivariate Analysis 00 (009) 305 3 = C q y y x x k k k= z q Γ [x k ( + z) + ξ k ] Γ [y ( + z) + η ], Re(z) 0, k= where C is a constant determined by E(Λ 0 ) =, x k = n, k =,..., q, y = qn, ξ k = (rk(z) + q rk(x) + k ), k =,..., rk(x), ξ rk(x)+k = (rk(z )+k ), k =,..., q rk(x), η = [rk(x)rk(z)+(q rk(x))rk(z )]. Therefore, based on the discussions in pp. 304 307 of Muirhead [3] or Box [], we immediately obtain Theorem 5. When the sphericity hypothesis (3) holds, and where P( log Λ u) = P(χ q(q+)/ u) + O(n ), P( ρ log Λ u) = P(χ q(q+)/ u) + O(n ), ρ = [q(q + ) ]n (A A ), [ ( rk(x) A = rk(x) (rk(z) + q rk(x))(rk(z) + q + ) + rk(x) + )] 3 [ ( + (q rk(x)) rk(z )(rk(z ) + q rk(x) + ) + (q rk(x)) 3 (q rk(x)) + )] q 6, A = [ (rk(x)rk(z) + (q rk(x))rk(z ) + ) ]. q 3 Acknowledgments This research was supported by the National Science Foundation (No. 07785 and 07600) of China. The author would like to express his thanks to the referees whose comments helped in improving his results. References [] V.M. Chinchilli, R.K. Elswick, A mixture of the MANOVA and GMANOVA models, Commun. Statist. Theory Methods 4 () (985) 3075 3089. [] T.U. Kollo, D. von Rosen, Advanced Multivariate Statistics with Matrices: Mathematics and its Applications, Springer, Dordrecht, New York, 005. [3] R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 98. [4] P. Bai, Exact distribution of MLE of covariance matrix in a GMANOVA MANOVA model, Sci. China Ser. A: Math. 48 () (005) 597 608. [5] M. Okamato, Distinctness of the eigenvalues of a quadratic form in a multivariate sample, Ann. Statist. 4 () (973) 763 765. [6] J.R. Schott, Matrix Analysis for Statistics, second edn, Wiley, New York, 005. [7] P.J. Bickel, K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Holden-day, Inc., San Francisco, 977. [8] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 999. [9] A. Erdélyi (Ed.), Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 953. [0] A.W. Davis, On the differential equation for Meijer s G p,0 p,p function, and further tables of Wilks s likelihood ratio criterion, Biometrika 3 (66) (979) 59 53. [] G.E.P. Box, A general distribution theory for a class of likelihood criteria, Biometrika 3 (36) (949) 37 346.