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.

Introduction Knowledge of tumor growth is important in the planning and evaluation of screening programs, clinical trials, and epidemiological studies. Studies of tumor growth rates in humans are usually based on small and selected samples. In the present study based on the Norwegian Breast Cancer Screening Program, tumor growth was estimated from a large population using a new estimating procedure/model.

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We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and un-observed heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.

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