How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?

Abstract

We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests when the correlation matrix is otherwise unrestricted. In each setting we construct a $q$-indexed family of finite Gaussian models whose FDR divided by $q$ diverges as $q\downarrow 0$, disproving any universal multiplicative FDR bound. For two-sided tests, the supremum over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ${\ell }_{=}(q)>q$ satisfying \[ \ell_{=}(q)=\frac{q\sqrt{\log(1/q)}}{2\sqrtπ}+c_\ell q+o(q), \qquad c_\ell=0.6492828\ldots. \] For the one-sided hypotheses $H_i:{\theta }_i\le 0$, a sign-reversed one-common-factor construction gives the stronger explicit lower bound ${\ell }_{\le }(q)>q$, with \[ \ell_{\le}(q)=\frac{q\sqrt{\log(1/q)}}{\sqrtπ} +\frac q2+o(q). \] Finally, we prove an $O\{q\sqrt{\log (1/q)}\}$ upper bound for the two-sided {one-common-factor} class and the matching upper bound $q\sqrt{\log (1/q)}/\sqrt{\pi }+O(q)$ for the one-sided one-common-factor class.

70 pages, changed the paper title, added lower and upper bounds for one-sided Gaussian tests