## Electronic Journal of Statistics

### False discovery rate control under Archimedean copula

#### Abstract

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.

#### Article information

Source
Electron. J. Statist., Volume 8, Number 2 (2014), 2207-2241.

Dates
First available in Project Euclid: 29 October 2014

https://projecteuclid.org/euclid.ejs/1414588192

Digital Object Identifier
doi:10.1214/14-EJS950

Mathematical Reviews number (MathSciNet)
MR3273624

Zentralblatt MATH identifier
1305.62269

Subjects
Primary: 62J15: Paired and multiple comparisons 62F05: Asymptotic properties of tests
Secondary: 62F03: Hypothesis testing

#### Citation

Bodnar, Taras; Dickhaus, Thorsten. False discovery rate control under Archimedean copula. Electron. J. Statist. 8 (2014), no. 2, 2207--2241. doi:10.1214/14-EJS950. https://projecteuclid.org/euclid.ejs/1414588192

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