Methods for improvement in estimation of a normal mean matrix

Abstract

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron–Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron–Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James–Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.