Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: A unified approach

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 FOR. Storey introduced a point estimate of the FOR for fixed significance regions. The former approach conservatively controls the FOR at a fixed predetermined level, and the latter provides a conservatively biased estimate of the FOR 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 FOR controlling procedures. In the asymptotic setting, we also show that the point estimates can be used to estimate the FOR conservatively over all significance regions simultaneously, which is equivalent to controlling the FOR at all levels simultaneously. The main tool that we use is to translate existing FOR 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.