• Bonferroni test;
  • Empirical distribution function;
  • Multiple comparisons;
  • Multiple-hypotheses testing;
  • Multiple level;
  • Multiple-test procedure;
  • Oracle procedure;
  • Plug-in estimator;
  • Step-down test

Summary.  Estimation of the number or proportion of true null hypotheses in multiple-testing problems has become an interesting area of research. The first important work in this field was performed by Schweder and Spjøtvoll. Among others, they proposed to use plug-in estimates for the proportion of true null hypotheses in multiple-test procedures to improve the power. We investigate the problem of controlling the familywise error rate FWER when such estimators are used as plug-in estimators in single-step or step-down multiple-test procedures. First we investigate the case of independent p-values under the null hypotheses and show that a suitable choice of plug-in estimates leads to control of FWER in single-step procedures. We also investigate the power and study the asymptotic behaviour of the number of false rejections. Although step-down procedures are more difficult to handle we briefly consider a possible solution to this problem. Anyhow, plug-in step-down procedures are not recommended here. For dependent p-values we derive a condition for asymptotic control of FWER and provide some simulations with respect to FWER and power for various models and hypotheses.