Admissibility and Complete Classes for False Discovery Rate Control with E-values

2026-07-15

AI Summary

This study investigates the admissibility and complete class problems for false discovery rate (FDR) control procedures within the e-value framework. Drawing on statistical decision theory, it introduces strong and weak dominance relations to establish, for the first time, a theoretical foundation for admissibility in e-value-based multiple testing with FDR control. The main contributions include proving that every step-down procedure is strongly dominated by some weighted average eBH procedure; demonstrating that weighted average eBH procedures without constant terms are admissible at any FDR level; and showing that, under symmetry, this class of procedures forms a complete class, with its members being maximal only when the FDR threshold is sufficiently small—thereby establishing their structural optimality.

Abstract

The false discovery rate (FDR) is the most widely used error metric in modern multiple testing. We provide the first comprehensive analysis of the admissibility of e-value-based procedures with FDR control. We consider both simultaneous and point procedures and introduce strong and weak notions of dominance. We show that every simultaneous procedure is strongly, and hence weakly, dominated by an admissible weighted-mean closed e-Benjamini-Hochberg ($\overline{\mathrm{eBH}}$) procedure, so weighted-mean $\overline{\mathrm{eBH}}$ procedures form a complete class. Moreover, every constant-free weighted-mean $\overline{\mathrm{eBH}}$ procedure is admissible at every level. Within the symmetric class, the usual mean $\overline{\mathrm{eBH}}$ procedure is the largest element if and only if the FDR level is small enough; otherwise this class has no largest element. We also obtain results on the admissibility of symmetric $\overline{\mathrm{eBH}}$ procedures with non-zero constant terms, and give guidance on the choice of the constant terms. Point e-testing procedures have a parallel theory for admissibility, where point weighted-mean $\overline{\mathrm{eBH}}$ procedures form a complete class. These results highlight the central role of weighted-mean $\overline{\mathrm{eBH}}$ procedures in multiple testing.

Problem

• false discovery rate
• e-values
• admissibility
• multiple testing
• complete class

Innovation

• e-values
• false discovery rate
• admissibility
• complete class
• weighted-mean eBH