Weighted shrinkage estimators of normal mean matrices and dominance properties

Abstract

In the estimation of the mean matrix in a multivariate normal distribution, the Efron–Morris estimator and the James–Stein estimator are two well-known minimax procedures, where the former is matricial shrinkage and the latter is scalar shrinkage. The methods for combining the two estimators with random weight functions are addressed. For deriving weight functions, the paper suggests the two methods. One is the minimization of a part of the unbiased estimator of the risk function, and the other is the empirical Bayes approach. The resulting weighted shrinkage estimators are shown to be minimax, and the extension to the case of an unknown covariance matrix is developed.