Dependency and false discovery rate: Asymptotics

TL;DR

This paper analyzes the asymptotic behavior of the false discovery rate and expected error rate in multiple hypothesis testing under dependence, providing explicit formulas and exploring models like equi-correlated normals and t-variables.

Contribution

It introduces a detailed asymptotic analysis of FDR and EER under dependence, linking Simes' rejection curve with empirical distributions, and computes limits for specific models.

Findings

Explicit formulas for FDR and EER in dependent models

Asymptotic behavior of FDR under dependence and independence

Numerical validation of theoretical results

Abstract

Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes' rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence.