Risk Estimate Optimality of James-Stein Estimators

Abstract

This note extends a result of Efron and Morris on domination of the maximum likelihood estimator for the mean of a multivariate normal distribution. We show that this result and our extension follow from a certain differential inequality. In a certain class of estimators having a unique unbiased estimator for the quadratic risk we find necessary and sufficient conditions for risk estimate dominance of a particular set of estimators. We show that, in the sense of risk estimates, these conditions imply that there are no estimators in this class which dominate the James-Stein or truncated James-Stein estimators.