On a generalized false discovery rate

TL;DR

This paper introduces new procedures for controlling the $k$-FDR in multiple testing, providing theoretical guarantees, demonstrating superior power through simulations, and applying the methods to real data.

Contribution

The paper develops new $k$-FDR controlling procedures with proven theoretical guarantees and improved power over existing methods.

Findings

Proposed methods control $k$-FDR under weaker conditions.

Numerical simulations show higher power compared to existing methods.

Application to real data demonstrates practical utility.

Abstract

The concept of $k$-FWER has received much attention lately as an appropriate error rate for multiple testing when one seeks to control at least $k$ false rejections, for some fixed $k\ge 1$. A less conservative notion, the $k$-FDR, has been introduced very recently by Sarkar [Ann. Statist. 34 (2006) 394--415], generalizing the false discovery rate of Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289--300]. In this article, we bring newer insight to the $k$-FDR considering a mixture model involving independent $p$-values before motivating the developments of some new procedures that control it. We prove the $k$-FDR control of the proposed methods under a slightly weaker condition than in the mixture model. We provide numerical evidence of the proposed methods' superior power performance over some $k$-FWER and $k$-FDR methods. Finally, we apply our methods to a real data set.