Minimaxity of the Stein risk-minimization estimator for a normal mean matrix

This paper addresses the Stein conjecture in the simultaneous estimation of a matrix mean of a multivariate normal distribution with a known covariance matrix. Stein (1973) derived an unbiased estimator of a risk function for orthogonally equivariant estimators and considered to isotonize the estimator which minimizes the main part of the unbiased risk-estimator. We call it the Stein risk-minimization estimator (RM) in this paper. Although the Stein RM estimator has been recognized as an excellent procedure with a nice risk-performance, it has a complicated form based on the isotonizing algorithm, and no analytical properties such as minimaxity have been shown. The aim of this paper is to fix this conjecture in lower dimensional cases, that is, the minimaxity of the Stein RM estimator is established for the two and three dimensions.