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In this paper, we construct a point estimator when assuming unequal and unknown variances by using the empirical Bayes approach in the classical normal mean problem. The proposed estimator shrinks both means and variances, and is thus called the double shrinkage estimator. Extensive numerical studies indicate that the double shrinkage estimator has lower Bayes risk than the estimator which shrinks the means alone, and the naive estimator which has no shrinkage at all. We further use a spike-in data set to assess different estimating procedures. It turns out that our proposed estimator performs the best and is thus strongly recommended for applications. AMS 2000 subject classifications: Primary 60K35, 60K35; secondary 60K35.

Double shrinkage empirical Bayesian estimation for unknown and unequal variances