False Discovery Rate Control for Sequential Selection Procedures, with Application to the Lasso

We consider a hypothesis testing scenario where we have a p-value p_j for each of a set of hypotheses H_1, H_2, ..., H_m, and these hypotheses must be rejected in a sequential manner. Because of the sequential setup, the standard approach of Benjamini and Hochberg cannot be applied. We propose two novel procedures called ForwardStop and StrongStop that are closely related to the Benjamini-Hochberg procedure, and prove that their False Discovery Rate is controlled at a pre-specified level in the sequential layout. This paper is motivated by recent work of Lockhart et al. deriving p-values for forward adaptive regression in the lasso framework. We apply ForwardStop and StrongStop to the lasso, and also propose two specialized procedures for it.