False discovery rate control under Archimedean copula. (English) Zbl 1305.62269

Summary: We are concerned with the false discovery rate (FDR) of the linear step-up test \(\varphi^{\mathrm{LSU}}\) considered by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)]. It is well known that \(\varphi^{\mathrm{LSU}}\) 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 \(\varphi^{\mathrm{LSU}}\) 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 \(\varphi^{\mathrm{LSU}}\) by pre-estimating the copula parameter and adjusting \(q\). Based on the lower bound, a sufficient condition is obtained under which the FDR of \(\varphi^{\mathrm{LSU}}\) 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.


62J15 Paired and multiple comparisons; multiple testing
62F05 Asymptotic properties of parametric tests
62F03 Parametric hypothesis testing


Zbl 0809.62014


Full Text: DOI arXiv Euclid


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