RISK RATIO AND MINIMAXITY IN ESTIMATING THE MULTIVARIATE NORMAL-MEAN WITH UNKNOWN-VARIANCE

The problem of estimating the mean of a multivariate normal distributi on with a common unknown variance is discussed. Risk functions under s quared error loss and their lower bound for a widely applied class of empirical Bayes estimators are given. The conditions for reaching the lower bound are derived by examining the risk ratio in terms of the ma ximum likelihood estimate when the dimension is large. When the dimens ion is moderate, a necessary and sufficient condition for an estimator to dominate the maximum likelihood estimate is found, establishing a minimaxity result. The James-Stein estimator, its positive-part versio n, the modal estimator and the hierarchical Bayes estimator are derive d and investigated.