February 2024 Rank-based indices for testing independence between two high-dimensional vectors
Yeqing Zhou, Kai Xu, Liping Zhu, Runze Li
Author Affiliations +
Ann. Statist. 52(1): 184-206 (February 2024). DOI: 10.1214/23-AOS2339


To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding’s D, Blum–Kiefer–Rosenblatt’s R and Bergsma–Dassios–Yanagimoto’s τ. Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily-tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high-dimensional settings. We establish explicit relationships between D, R, τ and Pearson’s correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative: (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on D, τ and R are sorted in a descending order.

Funding Statement

Yeqing Zhou is supported by National Natural Science Foundation of China (12001405), Natural Science Foundation of Shanghai (23ZR1469000) and Fundamental Research Funds for the Central Universities (22120210557).
Kai Xu is supported by National Natural Science Foundation of China (12271005, 11901006), Natural Science Foundation of Anhui Province (2308085Y06, 1908085QA06) and Young Scholars Program of Anhui Province (2023).
Liping Zhu is supported by Renmin University of China (22XNA026), National Natural Science Foundation of China (12225113, 12171477).
Runze Li’s research is partially supported by National Institute of Health (NIH) grants R01AI170249 and R01AI136664. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.


The authors thank the Editor, the Associate Editor and the anonymous reviewers for their constructive comments, which have led to a dramatic improvement of the earlier version of this article. The authors are also very grateful to Dr. Hongjian Shi for providing us with the R codes of Shi et al. (2022). The authors have contributed equally to this paper. All correspondence should be addressed to Liping Zhu (the corresponding author) at zhu.liping@ruc.edu.cn.


Download Citation

Yeqing Zhou. Kai Xu. Liping Zhu. Runze Li. "Rank-based indices for testing independence between two high-dimensional vectors." Ann. Statist. 52 (1) 184 - 206, February 2024. https://doi.org/10.1214/23-AOS2339


Received: 1 May 2022; Revised: 1 August 2023; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718412
Digital Object Identifier: 10.1214/23-AOS2339

Primary: 62G10
Secondary: 62G20

Keywords: Bergsma–Dassios–Yanagimoto’s τ∗ , Blum–Kiefer–Rosenblatt’s R , Degenerate U-statistics , Hoeffding’s D

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 1 • February 2024
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