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In the multiple testing context, a challenging problem is the estimation ofthe proportion $\pi_0$ of true-null hypotheses. A large number of estimators ofthis quantity rely on identifiability assumptions that either appear to beviolated on real data, or may be at least relaxed. Under independence, wepropose an estimator $\hat{\pi}_0$ based on density estimation using bothhistograms and cross-validation. Due to the strong connection between the falsediscovery rate (FDR) and $\pi_0$, many multiple testing procedures (MTP)designed to control the FDR may be improved by introducing an estimator of$\pi_0$. We provide an example of such an improvement (plug-in MTP) based onthe procedure of Benjamini and Hochberg. Asymptotic optimality results may bederived for both $\hat{\pi}_0$ and the resulting plug-in procedure. The latterensures the desired asymptotic control of the FDR, while it is more powerfulthan the BH-procedure. Finally, we compare our estimator of $\pi_0$ with otherwidespread estimators in a wide range of simulations. We obtain better resultsthan other tested methods in terms of mean square error (MSE) of the proposedestimator. Finally, both asymptotic optimality results and the interest intightly estimating $\pi_0$ are confirmed (empirically) by results obtained withthe plug-in MTP.

A leave-p-out based estimation of the proportion of null hypotheses