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

Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean