  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.
