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

摘要

For a GMANOVA-MANOVA model with normal error: Y = XB_1Z_1~T + B_2Z_2~T + E, E ～ N_(q x n)(0, I_n directX Σ), the present paper is devoted to the study of distribution of MLE, Σ, of covariance matrix Σ. The main results obtained are stated as follows: (1) When rk(Z) -rk(Z_2) ≥ q-rk(X), the exact distribution of Σ is derived, where Z = (Z_1,Z_2), rk(A) denotes the rank of matrix A. (2) The exact distribution of |Σ| is gained. (3) It is proved that ntT{[Σ~(-1)-Σ~(-1)XM(M~TX~TΣ~(-1)XM)~(-1)M~TX~TΣ~(-1)]Σ} has χ_((q-rk)(X))(n-rk(Z_2)))~2 distribution, where M is the matrix whose columns are the standardized orthogonal eigenvectors corresponding to the nonzero eigenvalues of X~TΣ~(-1)X.