  With recent advances in high throughput technology, researchers often find
themselves running a large number of hypothesis tests (thousands+) and esti-
mating a large number of effect-sizes. Generally there is particular interest
in those effects estimated to be most extreme. Unfortunately naive estimates of
these effect-sizes (even after potentially accounting for multiplicity in a
testing procedure) can be severely biased. In this manuscript we explore this
bias from a frequentist perspective: we give a formal definition, and show that
an oracle estimator using this bias dominates the naive maximum likelihood
estimate. We give a resampling estimator to approximate this oracle, and show
that it works well on simulated data. We also connect this to ideas in
empirical Bayes.
