April 2024 The edge of discovery: Controlling the local false discovery rate at the margin
Jake A. Soloff, Daniel Xiang, William Fithian
Author Affiliations +
Ann. Statist. 52(2): 580-601 (April 2024). DOI: 10.1214/24-AOS2359


Despite the popularity of the false discovery rate (FDR) as an error control metric for large-scale multiple testing, its close Bayesian counterpart the local false discovery rate (lfdr), defined as the posterior probability that a particular null hypothesis is false, is a more directly relevant standard for justifying and interpreting individual rejections. However, the lfdr is difficult to work with in small samples, as the prior distribution is typically unknown. We propose a simple multiple testing procedure and prove that it controls the expectation of the maximum lfdr across all rejections; equivalently, it controls the probability that the rejection with the largest p-value is a false discovery. Our method operates without knowledge of the prior, assuming only that the p-value density is uniform under the null and decreasing under the alternative. We also show that our method asymptotically implements the oracle Bayes procedure for a weighted classification risk, optimally trading off between false positives and false negatives. We derive the limiting distribution of the attained maximum lfdr over the rejections, and the limiting empirical Bayes regret relative to the oracle procedure.

Funding Statement

Jake Soloff was supported by NSF Grant DMS-2023505 and by the Office of Naval Research under the Vannevar Bush Fellowship, grant N00014-21-1-2941.
William Fithian was supported by the NSF Grant DMS-1916220 and a Hellman Fellowship from Berkeley.


We are indebted to Lihua Lei for simplifying the proof of Lemma 2. We also thank Rina Foygel Barber, Stephen Bates, Aditya Guntuboyina, Michael I. Jordan, Peter McCullagh and Jim Pitman for insightful discussions.


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Jake A. Soloff. Daniel Xiang. William Fithian. "The edge of discovery: Controlling the local false discovery rate at the margin." Ann. Statist. 52 (2) 580 - 601, April 2024. https://doi.org/10.1214/24-AOS2359


Received: 1 July 2022; Revised: 1 January 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2359

Primary: 62C12 , 62J15

Keywords: Empirical Bayes , Grenander estimator , local false discovery rates , multiple testing

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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