Stopping time property of thresholds of Storey-type FDR procedures

TL;DR

This paper establishes that Storey-type FDR procedures have a stopping time property and are conservative under certain conditions, providing theoretical guarantees for multiple testing correction.

Contribution

It introduces the concept of regular estimators of true nulls and proves the stopping time property of Storey-type procedures regardless of p-value dependence or distribution.

Findings

Rejection threshold is a stopping time with respect to the p-value filtration.

Storey-type FDR estimator equals the target FDR level at the rejection threshold.

Procedures are conservative under independent, uniform null p-values.

Abstract

For multiple testing, we introduce Storey-type FDR procedures and the concept of "regular estimator of the proportion of true nulls". We show that the rejection threshold of a Storey-type FDR procedure is a stopping time with respect to the backward filtration generated by the p-values and that a Storey-type FDR estimator at this rejection threshold equals the pre-specified FDR level, when the estimator of the proportion of true nulls is regular. These results hold regardless of the dependence among or the types of distributions of the p-values. They directly imply that a Storey-type FDR procedure is conservative when the null p-values are independent and uniformly distributed.