A New Proof of FDR Control Based on Forward Filtration

For multiple testing problems, Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the family-wise error rate (FWER). Since then, researchers have provided many proofs to control the FDR under different assumptions. Storey et al. (2004) showed that the rejection threshold of a BH step-up procedure is a stopping time with respect to the reverse filtration generated by the p-values and proposed a new proof based on the martingale theory. Following this work, martingale methods have been widely used to establish FDR control in various settings, but have been primarily applied to reverse filtration only. However, forward filtration can be more amenable for generalized and adaptive FDR controlling procedures. In this paper, we present a new proof, based on forward filtration, for step-down FDR controlling procedures that start from small p-values and update the rejection regions as larger p-values are observed.