False discovery rate control under Archimedean copula

We are concerned with the false discovery rate (FDR) of the linear step-up test φLSU considered by Benjamini and Hochberg (1995). It is well known that φLSU controls the FDR at level m₀q/m if the joint distribution of p-values is multivariate totally positive of order $2$. In this, m denotes the total number of hypotheses, m₀ 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 φLSU 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 φLSU by pre-estimating the copula parameter and adjusting q. Based on the lower bound, a sufficient condition is obtained under which the FDR of φLSU is exactly equal to m₀q/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.