Optimal Rates and Tradeoffs in Multiple Testing

Multiple hypothesis testing is a central topic in statistics. However, despite abundant research on the false discovery rate (FDR) and the corresponding Type-II error concept known as the false nondiscovery rate (FNR), we do not yet have a fine-grained understanding of the fundamental limits of multiple testing. The main contribution of this study is to derive a precise nonasymptotic trade-off between the FNR and FDR for a variant of the generalized Gaussian sequence model. Our approach is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses n, including various sparse and dense regimes (with o(n) and O(n) signals). Moreover, we prove that the Benjamini-Hochberg and Barber-Candes algorithms are both rate-optimal up to constants across these regimes.