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Multiple hypothesis testing often involves composite nulls, i.e., nulls thatare associated with two or more distributions. In many cases, it is reasonableto assume that there is a prior distribution on the distributions despite it isunknown. When the number of distributions under true nulls is finite, we showthat under the above assumption, the false discover rate (FDR) can becontrolled using $p$-values computed under constraints imposed by the empiricaldistribution of the observations. Comparing to FDR control using $p$-valuesdefined as maximum significance level over all null distributions, the proposedFDR control can have substantially more power.

FDR control for multiple hypothesis testing on composite nulls