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We consider Empirical Bayes (EB) estimation in the normal means problem, whenthe standard deviations of the observations are not known precisely, butestimated with error -- which is almost always the case in practicalapplications. In classical statistics accounting for estimated standard errorsusually involves replacing a normal distribution with a $t$ distribution. Thissuggests approaching this problem by replacing the normal assumption with a $t$assumption, leading to an "EB $t$-means problem". Here we show that an approachalong these lines can indeed work, but only with some care. Indeed, a naiveapplication of this idea is flawed, and can perform poorly. We suggest how thisflaw can be remedied by a two-stage procedure, which first performs EBshrinkage estimation of the standard errors and then solves an EB $t$-meansproblem. We give numerical results illustrating the effectiveness of thisremedy.

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