Ensemble minimaxity of James-Stein estimators

This article considers the estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators that shrink more on coordinates with larger variances seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that certain James-Stein type estimators can be ensemble minimax, that is, minimax with respect to the ensemble risk considered in the empirical Bayes perspective of Efron and Morris.