Minimax estimation of a normal covariance matrix with the partial Iwasawa decomposition

Abstract

This paper addresses the problem of estimating the normal covariance matrix relative to the Stein loss. The partial Iwasawa decomposition is used to reduce the original estimation problem to simultaneous estimation for variances and means of some normal distributions. The variances and the means are closely related to, respectively, the diagonal and the below-diagonal elements of a lower triangular matrix which is made from the Cholesky decomposition of the covariance matrix. Shrinkage type procedures are proposed for improvements not only on the diagonal elements but also on the below-diagonal elements corresponding to the James and Stein minimax estimator of the covariance matrix.