April 2024 Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics
Patrick Bastian, Holger Dette, Johannes Heiny
Author Affiliations +
Ann. Statist. 52(2): 628-653 (April 2024). DOI: 10.1214/24-AOS2361


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.

Funding Statement

This work was partially supported by the DFG Research unit 5381 Mathematical Statistics in the Information Age, project number 460867398.


The authors would like to thank two referees and the Associate Editor for constructive comments, which led to a substantial improvement of an earlier version of this paper.


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Patrick Bastian. Holger Dette. Johannes Heiny. "Testing for practically significant dependencies in high dimensions via bootstrapping maxima of U-statistics." Ann. Statist. 52 (2) 628 - 653, April 2024. https://doi.org/10.1214/24-AOS2361


Received: 1 July 2023; Revised: 1 January 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2361

Primary: 62C20 , 62F40 , 62G10

Keywords: bootstrap , Gaussian approximation , Independence testing , Minimax optimality , relevant association , U-statistics

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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