James–Stein estimation problem for a multivariate normal random matrix and an improved estimator

In this paper, we provide the proof of nonexistence of the James–Stein estimator in the whole parameter space for normal random matrices, equivalently, for multivariate linear regression models, which solves the open problem raised by S.F. Arnold [1]. By introducing the concepts of left and right James–Stein estimators, we obtain the left James–Stein estimator of mean matrix and show that the left James–Stein estimator has minimaxity and optimality in terms of the Efron–Morris type modification. We construct a new minimax combination estimator with lower risk by absorbing the advantages of the left James–Stein estimator and the existing modified Stein estimator. Risk comparisons through finite sample simulation studies illustrate that the proposed combination estimator has a better performance, under the mean-squared error or l2 risk, compared with all existing estimators.