Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics

Abstract

This paper takes a different look on the problem of testing the mutual independence of the components of a high-dimensional vector. Instead of testing if all pairwise associations (e.g., all pairwise Kendall’s τ) between the components vanish, we are interested in the (null) hypothesis that all pairwise associations do not exceed a certain threshold in absolute value. The consideration of these hypotheses is motivated by the observation that in the high-dimensional regime, it is rare, and perhaps impossible, to have a null hypothesis that can be exactly modeled by assuming that all pairwise associations are precisely equal to zero.

The formulation of the null hypothesis as a composite hypothesis makes the problem of constructing tests nonstandard and in this paper we provide a solution for a broad class of dependence measures, which can be estimated by U-statistics. In particular, we develop an asymptotic and a bootstrap level α-test for the new hypotheses in the high-dimensional regime. We also prove that the new tests are minimax-optimal and investigate their finite sample properties by means of a small simulation study and a data example.