Empirical Bayes estimation of normal means, accounting for uncertainty in estimated standard errors
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We consider Empirical Bayes (EB) estimation in the normal means problem, when the standard deviations of the observations are not known precisely, but estimated with error -- which is almost always the case in practical applications. In classical statistics accounting for estimated standard errors usually involves replacing a normal distribution with a t distribution. This suggests approaching this problem by replacing the normal assumption with a t assumption, leading to an "EB t-means problem". Here we show that an approach along these lines can indeed work, but only with some care. Indeed, a naive application of this idea is flawed, and can perform poorly. We suggest how this flaw can be remedied by a two-stage procedure, which first performs EB shrinkage estimation of the standard errors and then solves an EB t-means problem. We give numerical results illustrating the effectiveness of this remedy.
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