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On eigenvalues of a high-dimensional Kendall’s rank correlation matrix with dependence

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Abstract

In this paper, we investigate the limiting spectral distribution of a high-dimensional Kendall’s rank correlation matrix. The underlying population is allowed to have a general dependence structure. The result no longer follows the generalized Marc̆enko-Pastur law, which is brand new. It is the first result on rank correlation matrices with dependence. As applications, we study Kendall’s rank correlation matrix for multivariate normal distributions with a general covariance matrix. From these results, we further gain insights into Kendall’s rank correlation matrix and its connections with the sample covariance/correlation matrix.

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Acknowledgements

Zeng Li’s work was supported by National Natural Science Foundation of China (Grant Nos. 12031005 and 12101292). Cheng Wang’s work was supported by National Natural Science Foundation of China (Grant No. 12031005) and Natural Science Foundation of Shanghai (Grant No. 21ZR1432900). Qinwen Wang’s work was supported by National Natural Science Foundation of China (Grant No. 12171099). The authors contributed equally to this work and are listed in alphabetical order. The authors thank the reviewers for their insightful comments.

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Authors and Affiliations

  1. Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen, 518055, China

    Zeng Li

  2. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Cheng Wang

  3. Key Laboratory of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, 200240, China

    Cheng Wang

  4. School of Data Science, Fudan University, Shanghai, 200433, China

    Qinwen Wang

Authors
  1. Zeng Li
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  2. Cheng Wang
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  3. Qinwen Wang
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Correspondence to Cheng Wang.

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Li, Z., Wang, C. & Wang, Q. On eigenvalues of a high-dimensional Kendall’s rank correlation matrix with dependence. Sci. China Math. 66, 2615–2640 (2023). https://doi.org/10.1007/s11425-022-2031-2

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