Improved Gaussian mean matrix estimators in high-dimensional data

Abstract

In this paper, we introduce a class of improved estimators for the mean parameter matrix of a multivariate normal distribution with an unknown variance–covariance matrix. In particular, the main results of Chételat and Wells (2012) are established in their full generalities and we provide the corrected version of their Theorem 2. Specifically, we generalize the existing results in three ways. First, we consider a parametric estimation problem which encloses as a special case the one about the vector parameter. Second, we propose a class of James–Stein matrix estimators and, we establish a necessary and a sufficient condition for any member of the proposed class to have a finite risk function. Third, we present the conditions for the proposed class of estimators to dominate the maximum likelihood estimator. On the top of these interesting contributions, the additional novelty consists in the fact that, we extend the methods suitable for the vector parameter case and the derived results hold in the classical case as well as in the context of high and ultra-high dimensional data.