A unified approach to estimating a normal mean matrix in high and low dimensions

This paper addresses the problem of estimating the normal mean matrix with an unknown covariance matrix. Motivated by an empirical Bayes method, we suggest a unified form of the Efron–Morris type estimators based on the Moore–Penrose inverse. This form not only can be defined for any dimension and any sample size, but also can contain the Efron–Morris type or Baranchik type estimators suggested so far in the literature. Also, the unified form suggests a general class of shrinkage estimators. For shrinkage estimators within the general class, a unified expression of unbiased estimators of the risk functions is derived regardless of the dimension of covariance matrix and the size of the mean matrix. An analytical dominance result is provided for a positive-part rule of the shrinkage estimators.