Double shrinkage empirical Bayesian estimation for unknown and unequal variances

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.

Keywords

James–Stein estimator, lognormal model, loss function

2010 Mathematics Subject Classification

60K35

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Published 1 January 2010