  We consider the problem of estimating the proportion $\theta$ of true null
hypotheses in a multiple testing context. The setup is classically modeled
through a semiparametric mixture with two components: a uniform distribution on
interval $[0,1]$ with prior probability $\theta$ and a nonparametric density
$f$. We discuss asymptotic efficiency results and establish that two different
cases occur whether $f$ vanishes on a set with non null Lebesgue measure or
not. In the first case, we exhibit estimators converging at parametric rate,
compute the optimal asymptotic variance and conjecture that no estimator is
asymptotically efficient (i.e. attains the optimal asymptotic variance). In the
second case, we prove that the quadratic risk of any estimator does not
converge at parametric rate. We illustrate those results on simulated data.
