Self-consistent multiple testing procedures
Gilles Blanchard and Étienne Roquain
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.