On efficient estimators of the proportion of true null hypotheses in a multiple testing setup

van Hanh Nguyen (1, 2) , Catherine Matias (1)

1 SG - Laboratoire Statistique et Génome
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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 (\emph{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.

Mots clés: information bound; Asymptotic efficiency; false discovery rate; efficient score; multiple testing; $p$-values; semiparametric model; semiparametric model.

Domaines: Applications [stat.AP]

van Hanh Nguyen, Catherine Matias. On efficient estimators of the proportion of true null hypotheses in a multiple testing setup. Scandinavian Journal of Statistics, 2014, 41 (4), pp.1167-1194. ⟨10.1111/sjos.12091⟩. ⟨hal-00647082v3⟩