Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean

The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical Bayes estimators. By taking a different approach, that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent. To demonstrate the merits of the proposed methods in real applications, we analyze the baseball data used in Brown (2008), where the group-linear methods achieved the prediction error of the best nonparametric estimates that have been applied to the dataset, and significantly lower error than other parametric and semi-parametric empirical Bayes estimators.