Dynamic adaptive procedures that control the false discovery rate

In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level $\pi_0\alpha$, where $\pi_0$ is the proportion of true null hypotheses and $\alpha$ is the target FDR level. Adaptive procedures can improve power by incorporating estimates of $\pi_0$, which typically rely on a tuning parameter. Fixed adaptive procedures set their tuning parameters before seeing the data and can be shown to control the FDR in finite samples. We develop theoretical results for dynamic adaptive procedures whose tuning parameters are determined by the data. We show that, if the tuning parameter is chosen according to a left-to-right stopping time rule, the corresponding dynamic adaptive procedure controls the FDR in finite samples. Examples include the recently proposed right-boundary procedure and the widely used lowest-slope procedure, among others. Simulation results show that the right-boundary procedure is more powerful than other dynamic adaptive procedures under independence and mild dependence conditions.