Discoveries / probability-statistics

Universal Multiplicative FDR Bound for Benjamini-Hochberg

The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q)}/(2\sqrt{\pi}) + 0.6493 q + o(q)$.