December 2020 High-dimensional consistent independence testing with maxima of rank correlations
Mathias Drton, Fang Han, Hongjian Shi
Ann. Statist. 48(6): 3206-3227 (December 2020). DOI: 10.1214/19-AOS1926


Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting nonlinear, nonmonotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence. Built upon a newly developed Cramér-type moderate deviation theorem for degenerate U-statistics, our results cover a variety of rank correlations including Hoeffding’s $D$, Blum–Kiefer–Rosenblatt’s $R$ and Bergsma–Dassios–Yanagimoto’s $\tau^{*}$. The proposed tests are distribution-free in the class of multivariate distributions with continuous margins, implementable without the need for permutation, and are shown to be rate-optimal against sparse alternatives under the Gaussian copula model. As a by-product of the study, we reveal an identity between the aforementioned three rank correlation statistics, and hence make a step towards proving a conjecture of Bergsma and Dassios.


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Mathias Drton. Fang Han. Hongjian Shi. "High-dimensional consistent independence testing with maxima of rank correlations." Ann. Statist. 48 (6) 3206 - 3227, December 2020.


Received: 1 May 2019; Revised: 1 November 2019; Published: December 2020
First available in Project Euclid: 11 December 2020

Digital Object Identifier: 10.1214/19-AOS1926

Primary: 62G10

Keywords: Degenerate U-statistics , extreme value distribution , Independence test , maximum-type test , rank statistics , rate-optimality

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 6 • December 2020
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