  We study a stylized multiple testing problem where the test statistics are
independent and assumed to have the same distribution under their respective
null hypotheses. We first show that, in the normal means model where the test
statistics are normal Z-scores, the well-known method of (Benjamini and
Hochberg, 1995) is optimal in some asymptotic sense. We then show that this is
also the case of a recent distribution-free method proposed by Foygel-Barber
and Cand\`es (2015). The method is distribution-free in the sense that it is
agnostic to the null distribution - it only requires that the null distribution
be symmetric. We extend these optimality results to other location models with
a base distribution having fast-decaying tails.
