Ridge-type linear shrinkage estimation of the mean matrix of a high-dimensional normal distribution

Abstract

The estimation of the mean matrix of the multivariate normal distribution is addressed in the high dimensional setting. Efron–Morris-type linear shrinkage estimators with ridge modification for the precision matrix instead of the Moore–Penrose generalized inverse are considered, and the weights in the ridge-type linear shrinkage estimators are estimated in terms of minimizing the Stein unbiased risk estimators under the quadratic loss. It is shown that the ridge-type linear shrinkage estimators with the estimated weights are minimax, and that the estimated weights and the loss function with these estimated weights are asymptotically equal to the optimal counterparts in the Bayesian model with high dimension by using the random matrix theory. The performance of the ridge-type linear shrinkage estimators is numerically compared with the existing estimators including the Efron–Morris and James–Stein estimators.