Combining coordinates in simultaneous estimation of normal means

The problem of combining coordinates in Stein-type estimators, when simultaneously estimating normal means, is considered. The question of deciding whether to use all coordinates in one combined shrinkage estimator or to separate into groups and use separate shrinkage estimators on each group is considered. A Bayesian viewpoint is (of necessity) taken, and it is shown that the ‘combined’ estimator is, somewhat surprisingly, often superior.