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.