On the Maximum Likelihood Estimation of a Covariance Matrix

Abstract

For a multivariate normal set-up, it is well known that themaximumlikelihood estimator (MLE) of covariance matrix is neither admissible nor minimax under the Stein loss function. In this paper, we reveal that the MLE based on the Iwasawa parameterization leads to minimaxity with respect to the Stein loss function. Furthermore, a novel class of loss functions is proposed so that the minimum risks of the MLEs are identical in different coordinate systems, Cholesky parameterization and full Iwasawa parameterization. In other words, the MLEs based on these two different parameterizations are characterized by the property of minimaxity, without a Stein paradox. The application of our novel method to the high-dimensional covariance matrix problem is also discussed.