We consider the problem of estimating the proportion of true null hypotheses, "π" 0, in a multiple-hypothesis set-up. The tests are based on observed "p"-values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the "p"-value density, restricting to decreasing and convex decreasing densities. The estimators of "π" 0 are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence. Copyright 2005 Royal Statistical Society.
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