  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.
