Steinized Empirical Bayes Estimation for Heteroscedastic Data

Consider the problem of estimating normal means from independent observations with known variances, possibly different from each other. Suppose that a second-level normal model is specified on the unknown means, with the prior means depending on a vector of covariates and the prior variances constant. For this two-level normal model, existing empirical Bayes methods are constructed from the Bayes rule with the prior parameters selected either by maximum likelihood or moment equations or by minimizing Stein’s unbiased risk estimate. Such methods tend to deteriorate, sometimes substantially, when the second-level model is misspecified. We develop a Steinized empirical Bayes approach for improving the robustness to misspecification of the second-level model, while preserving the effectiveness in risk reduction when the second-level model is appropriate in capturing the unknown means. The proposed methods are constructed from a minimax Bayes estimator or, interpreted by its form, a Steinized Bayes estimator, which is not only globally minimax but also achieves close to the minimum Bayes risk over a scale class of normal priors including the specified prior. The prior parameters are then estimated by standard moment methods. We provide formal results showing that the proposed methods yield no greater asymptotic risks than existing methods using the same estimates of prior parameters, but without requiring the second-level model to be correct. We present both an application for predicting baseball batting averages and two simulation studies to demonstrate the practical advantage of the proposed methods.