False Discovery Rate Control under Archimedean Copula

TL;DR

This paper investigates the control of false discovery rate (FDR) in multiple hypothesis testing under Archimedean copula models, deriving bounds and conditions for FDR accuracy, and enhancing test power through copula parameter estimation.

Contribution

It introduces sharper FDR bounds for the linear step-up test under Archimedean copulas and proposes methods to improve test power by estimating copula parameters.

Findings

Derived sharper upper bounds for FDR under Archimedean copulas.

Established conditions for FDR equality to the nominal level.

Connected exchangeable p-values with Archimedean copula models.

Abstract

We are considered with the false discovery rate (FDR) of the linear step-up test $\varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $\varphi^{LSU}$ controls the FDR at level $m_0 q / m$ if the joint distribution of $p$-values is multivariate totally positive of order 2. In this, $m$ denotes the total number of hypotheses, $m_0$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $\varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\varphi^{LSU}$ is exactly equal to $m_0 q / m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel copulae.