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Block-diagonal test for high-dimensional covariance matrices

Author

Listed:
  • Jiayu Lai

    (Northeast Normal University)

  • Xiaoyi Wang

    (Beijing Normal University)

  • Kaige Zhao

    (Northeast Normal University)

  • Shurong Zheng

    (Northeast Normal University)

Abstract

The structure testing of a high-dimensional covariance matrix plays an important role in financial stock analyses, genetic series analyses, and many other fields. Testing that the covariance matrix is block-diagonal under the high-dimensional setting is the main focus of this paper. Several test procedures that rely on normality assumptions, two-diagonal block assumptions, or sub-block dimensionality assumptions have been proposed to tackle this problem. To relax these assumptions, we develop a test framework based on U-statistics, and the asymptotic distributions of the U-statistics are established under the null and local alternative hypotheses. Moreover, a test approach is developed for alternatives with different sparsity levels. Finally, both a simulation study and real data analysis demonstrate the performance of our proposed methods.

Suggested Citation

  • Jiayu Lai & Xiaoyi Wang & Kaige Zhao & Shurong Zheng, 2023. "Block-diagonal test for high-dimensional covariance matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 447-466, March.
    • Handle: RePEc:spr:testjl:v:32:y:2023:i:1:d:10.1007_s11749-022-00842-x
    DOI: 10.1007/s11749-022-00842-x
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    References listed on IDEAS

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    1. Tatjana Pavlenko & Anders Björkström & Annika Tillander, 2012. "Covariance structure approximation via gLasso in high-dimensional supervised classification," Journal of Applied Statistics, Taylor & Francis Journals, vol. 39(8), pages 1643-1666, January.
    2. Yongcheng Qi & Fang Wang & Lin Zhang, 2019. "Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 911-946, August.
    3. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    4. Xiao, Han & Wu, Wei Biao, 2013. "Asymptotic theory for maximum deviations of sample covariance matrix estimates," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2899-2920.
    5. Srivastava, Muni S. & Reid, N., 2012. "Testing the structure of the covariance matrix with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 156-171.
    6. James R. Schott, 2005. "Testing for complete independence in high dimensions," Biometrika, Biometrika Trust, vol. 92(4), pages 951-956, December.
    7. Lin, Zhengyan & Xiang, Yanbiao, 2008. "A hypothesis test for independence of sets of variates in high dimensions," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2939-2946, December.
    8. Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2019. "Testing for independence of large dimensional vectors," MPRA Paper 97997, University Library of Munich, Germany, revised May 2019.
    9. Weiming Li & Jianfeng Yao, 2018. "On structure testing for component covariance matrices of a high dimensional mixture," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 293-318, March.
    10. Yata, Kazuyoshi & Aoshima, Makoto, 2016. "High-dimensional inference on covariance structures via the extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 151-166.
    11. Emilie Devijver & Mélina Gallopin, 2018. "Block-Diagonal Covariance Selection for High-Dimensional Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 306-314, January.
    12. Yamada, Yuki & Hyodo, Masashi & Nishiyama, Takahiro, 2017. "Testing block-diagonal covariance structure for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 305-316.
    13. Chen, Song Xi & Zhang, Li-Xin & Zhong, Ping-Shou, 2010. "Tests for High-Dimensional Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 810-819.
    14. Dudoit S. & Fridlyand J. & Speed T. P, 2002. "Comparison of Discrimination Methods for the Classification of Tumors Using Gene Expression Data," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 77-87, March.
    15. Tiefeng Jiang & Yongcheng Qi, 2015. "Likelihood Ratio Tests for High-Dimensional Normal Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 988-1009, December.
    16. Xu, Kai & Hao, Xinxin, 2019. "A nonparametric test for block-diagonal covariance structure in high dimension and small samples," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 551-567.
    17. Masashi Hyodo & Nobumichi Shutoh & Takahiro Nishiyama & Tatjana Pavlenko, 2015. "Testing block-diagonal covariance structure for high-dimensional data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 460-482, November.
    18. Weiping Zhang & Baisuo Jin & Zhidong Bai, 2021. "Learning block structures in U-statistic-based matrices [Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis]," Biometrika, Biometrika Trust, vol. 108(4), pages 933-946.
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