The FDR-Linking Theorem
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This paper introduces the FDR-linking theorem, a novel technique for understanding non-asymptotic FDR control of the Benjamini--Hochberg (BH) procedure under arbitrary dependence of the p-values. This theorem offers a principled and flexible approach to linking all p-values and the null p-values from the FDR control perspective, suggesting a profound implication that, to a large extent, the FDR of the BH procedure relies mostly on the null p-values. To illustrate the use of this theorem, we propose a new type of dependence only concerning the null p-values, which, while strictly relaxing the state-of-the-art PRDS dependence (Benjamini and Yekutieli, 2001), ensures the FDR of the BH procedure below a level that is independent of the number of hypotheses. This level is, furthermore, shown to be optimal under this new dependence structure. Next, we present a concept referred to as FDR consistency that is weaker but more amenable than FDR control, and the FDR-linking theorem shows that FDR consistency is completely determined by the joint distribution of the null p-values, thereby reducing the analysis of this new concept to the global null case. Finally, this theorem is used to obtain a sharp FDR bound under arbitrary dependence, which improves the -correction FDR bound (Benjamini and Yekutieli, 2001) in certain regimes.
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