Strong Control, Conservative Point Estimation and Simultaneous Conservative Consistency of False Discovery Rates: A Unified Approach

TLDR

The false discovery rate (FDR) quantifies the expected proportion of false positives among rejected hypotheses, was introduced by Benjamini and Hochberg with a step‑up procedure that controls it, and later extended by Storey with a conservatively biased point estimate for fixed significance regions. This study demonstrates that the conservative control and point‑estimation approaches to FDR are essentially equivalent in both finite‑sample and asymptotic regimes. The authors translate existing FDR methods into empirical‑process‑based procedures. They find that FDR point estimates can define valid controlling procedures, enable conservative estimation across all significance regions simultaneously, and remain valid even under certain dependencies, thereby simplifying finite‑sample proofs and providing a unified asymptotic framework.

Abstract

Summary The false discovery rate (FDR) is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses. Benjamini and Hochberg introduced this quantity and proved that a particular step-up p-value method controls the FDR. Storey introduced a point estimate of the FDR for fixed significance regions. The former approach conservatively controls the FDR at a fixed predetermined level, and the latter provides a conservatively biased estimate of the FDR for a fixed predetermined significance region. In this work, we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent. In particular, the FDR point estimates can be used to define valid FDR controlling procedures. In the asymptotic setting, we also show that the point estimates can be used to estimate the FDR conservatively over all significance regions simultaneously, which is equivalent to controlling the FDR at all levels simultaneously. The main tool that we use is to translate existing FDR methods into procedures involving empirical processes. This simplifies finite sample proofs, provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence.