Estimating the proportion of true null hypotheses when the statistics are discrete

In high-dimensional testing problems π0, the proportion of null hypotheses that are true is an important parameter. For discrete test statistics, the P values come from a discrete distribution with finite support and the null distribution may depend on an ancillary statistic such as a table margin that varies among the test statistics. Methods for estimating π0 developed for continuous test statistics, which depend on a uniform or identical null distribution of P values, may not perform well when applied to discrete testing problems.