False Discovery Rate Control under Archimedean Copula

We prove that the linear step-up procedure $\vp^{LSU}$ considered by Benjamini and Hochberg (1995) controls the false discovery rate (FDR) in the case of dependent $p$-values whose dependency structure is determined by an Archimedean copula. In fact, the FDR of $\vp^{LSU}$ is under the assumption of an Archimedean $p$-value copula upper-bounded by the same constant as in the case of independent $p$-values. Namely, the upper bound is given by $m_0 q / m$, where $m$ denotes the total number of hypotheses, $m_0$ the number of true null hypotheses, and $q$ the nominal FDR level. Furthermore, we establish a sharper upper bound for the FDR of $\vp^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\vp^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\vp^{LSU}$ is exactly equal to $m_0 q / m$. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by proving that the dependency structure of the corresponding vector of $p$-values can always be expressed by an Archimedean copula. The theoretical results are applied to important copula families, including Clayton copulae and Gumbel copulae.