  Multiple testing with false discovery rate (FDR) control has been widely
conducted in the ``discrete paradigm" where p-values have discrete and
heterogeneous null distributions. However, in this scenario existing FDR
procedures often lose some power and may yield unreliable inference, and for
this scenario there does not seem to be an FDR procedure that partitions
hypotheses into groups, employs data-adaptive weights and is non-asymptotically
conservative. We propose a weighted FDR procedure for multiple testing in the
discrete paradigm that efficiently adapts to both the heterogeneity and
discreteness of p-value distributions. We theoretically justify the
non-asymptotic conservativeness of the weighted FDR procedure under
independence, and show via simulation studies that, for multiple testing based
on p-values of Binomial test or Fisher's exact test, it is more powerful than
six other procedures. The weighted FDR procedure is applied to a drug safety
study and a differential methylation study based on discrete data, where it
makes more discoveries than two existing methods.
