Empirical Bayes estimation of normal means, accounting for uncertainty in estimated standard errors

Abstract: 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.