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

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.