Open Access
2014 False discovery rate control under Archimedean copula
Taras Bodnar, Thorsten Dickhaus
Electron. J. Statist. 8(2): 2207-2241 (2014). DOI: 10.1214/14-EJS950


We are concerned 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-Hougaard copulae.


Download Citation

Taras Bodnar. Thorsten Dickhaus. "False discovery rate control under Archimedean copula." Electron. J. Statist. 8 (2) 2207 - 2241, 2014.


Published: 2014
First available in Project Euclid: 29 October 2014

zbMATH: 1305.62269
MathSciNet: MR3273624
Digital Object Identifier: 10.1214/14-EJS950

Primary: 62F05 , 62J15
Secondary: 62F03

Keywords: $P$-values , Clayton copula , exchangeability , Gumbel- Hougaard copula , linear step-up test , Multiple hypotheses testing

Rights: Copyright © 2014 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.8 • No. 2 • 2014
Back to Top