Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data

摘要

In this paper, we study the problem of estimating the covariance matrix Sigma and the precision matrix Omega (the inverse of the covariance matrix) in a star-shape model with missing data. By considering a type of Cholesky decomposition of the precision matrix Omega = Psi'Psi, where Psi is a lower triangular matrix with positive diagonal elements, we get the MLEs of the covariance matrix and precision matrix and prove that both of them are biased. Based on the MLEs, unbiased estimators of the covariance matrix and precision matrix are obtained. A special group G, which is a subgroup of the group consisting all lower triangular matrices, is introduced. By choosing the left invariant Haar measure on g as a prior, we obtain the closed forms of the best equivariant estimates of Omega under any of the Stein loss, the entropy loss, and the symmetric loss. Consequently, the MLE of the precision matrix (covariance matrix) is inadmissible under any of the above three loss functions. Some simulation results are given for illustration. (C) 2005 Elsevier Inc. All rights reserved.