Asymptotic Results on Adaptive False Discovery Rate Controlling Procedures Based on Kernel Estimators

TL;DR

This paper analyzes the asymptotic behavior of kernel-based plug-in procedures for FDR control, demonstrating increased power and tighter control as the number of hypotheses grows, with convergence rates depending on distribution regularity.

Contribution

It provides the first asymptotic analysis of kernel estimator-based FDR procedures, showing their improved power and control properties in large-sample settings.

Findings

Kernel-based procedures achieve tighter asymptotic FDR control.

They have a broader range of target levels with positive asymptotic power.

Convergence rates depend on the regularity of the p-value distribution.

Abstract

The False Discovery Rate (FDR) is a commonly used type I error rate in multiple testing problems. It is defined as the expected False Discovery Proportion (FDP), that is, the expected fraction of false positives among rejected hypotheses. When the hypotheses are independent, the Benjamini-Hochberg procedure achieves FDR control at any pre-specified level. By construction, FDR control offers no guarantee in terms of power, or type II error. A number of alternative procedures have been developed, including plug-in procedures that aim at gaining power by incorporating an estimate of the proportion of true null hypotheses. In this paper, we study the asymptotic behavior of a class of plug-in procedures based on kernel estimators of the density of the $p$-values, as the number $m$ of tested hypotheses grows to infinity. In a setting where the hypotheses tested are independent, we prove that these procedures are asymptotically more powerful in two respects: (i) a tighter asymptotic FDR control for any target FDR level and (ii) a broader range of target levels yielding positive asymptotic power. We also show that this increased asymptotic power comes at the price of slower, non-parametric convergence rates for the FDP. These rates are of the form $m^{-k/(2k+1)}$, where $k$ is determined by the regularity of the density of the $p$-value distribution, or, equivalently, of the test statistics distribution. These results are applied to one- and two-sided tests statistics for Gaussian and Laplace location models, and for the Student model.