Self-consistent multiple testing procedures

We study the control of the false discovery rate (FDR) for a general class of multiple testing procedures. We introduce a general condition, called ``self-consistency'', on the set of hypotheses rejected by the procedure, which we show is sufficient to ensure the control of the corresponding false discovery rate under various conditions on the distribution of the $p$-values. Maximizing the size of the rejected null hypotheses set under the constraint of self-consistency, we recover various step-up procedures. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: arbitrary measure of set size, $p$-value reweighting, new family of step-up procedures under unspecified $p$-value dependency. Our framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.