Exact distribution of MLE of covariance matrix in a GMANOVA-MANOVA model

Abstract

For a GMANOVA-MANOVA model with normal error: Y-XB ₁ Z ^T₁ +B ₂ Z ^T₂ +E, E∽N _q×n(0, I _n⊗∑), the present paper is devoted to the study of distribution of MLE, $\hat \Sigma$, of covariance matrix Σ. The main results obtained are stated as follows: (1) When rk(Z) ? rk(Z₂) ≥ q - rk(X), the exact distribution of $\hat \Sigma$ is derived, where Z = (Z1,Z2), rk(A) denotes the rank of matrix A. (2) The exact distribution of $\left| {\hat \Sigma } \right|$ is gained. (3) It is proved that $ntr\left\{ {\left[ {\Sigma ^{ - 1} - \Sigma ^{ - 1} XM(M^T X^T \Sigma ^{ - 1} - XM)^{ - 1} M^T X^T \Sigma ^{ - 1} } \right]\hat \Sigma } \right\}$ 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 X ^T Σ^-1 X.