For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtaine...For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.展开更多
This article presents a statistic for testing the sphericity in a GMANOVA- MANOVA model with normal error. It is shown that the null distribution of this statistic is beta and its nonnull distribution is given in seri...This article presents a statistic for testing the sphericity in a GMANOVA- MANOVA model with normal error. It is shown that the null distribution of this statistic is beta and its nonnull distribution is given in series form of beta distributions.展开更多
基金supported by the National Naural Science Foundation of China(Grant No.1026 1009)Mathematics Tianyuan Youth Foundation of China,
文摘For a GMANOVA-MANOVA model with normal error: Y = XB1Z1 T +B2Z2 T+ E, E- Nq×n(0, In (?) ∑), the present paper is devoted to the study of distribution of MLE, ∑, of covariance matrix ∑. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z2) ≥ q-rk(X), the exact distribution of ∑ is derived, where z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |∑| is gained. (3) It is proved that ntr{[S-1 - ∑-1XM(MTXT∑-1XM)-1MTXT∑-1]∑}has X2(q_rk(x))(n-rk(z2)) distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of XT∑-1X.
基金the National Natural Science Foundation of China (10761010, 10771185)the Mathematics Tianyuan Youth Foundation of China
文摘This article presents a statistic for testing the sphericity in a GMANOVA- MANOVA model with normal error. It is shown that the null distribution of this statistic is beta and its nonnull distribution is given in series form of beta distributions.
