RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS

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.

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Work Title RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS
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Open Access
Creators
  1. Yeqing Zhou
  2. Kai Xu
  3. Liping Zhu
  4. Runze Li
Keyword
  1. Bergsma-Dassios-Yanagimotos t*
  2. Blum-Kiefer-Rosenblatt's R
  3. Degenerate U-statistics
  4. Hoffding's D
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Annals of Statistics
Publication Date February 2024
Publisher Identifier (DOI)
  1. https://doi.org/10.1214/23-aos2339
Deposited October 07, 2024

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Version 1
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  • Created
  • Added AOS2339.pdf
  • Added Creator Yeqing Zhou
  • Added Creator Kai Xu
  • Added Creator Liping Zhu
  • Added Creator Runze Li
  • Published
  • Updated
  • Updated Keyword, Publication Date Show Changes
    Keyword
    • Bergsma-Dassios-Yanagimotos t*, Blum-Kiefer-Rosenblatt's R, Degenerate U-statistics, Hoffding's D
    Publication Date
    • 2024-02-01
    • 2024-02