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Limiting distributions of likelihood ratio test for independence of components for high-dimensional normal vectors

Author

Listed:
  • Yongcheng Qi

    (University of Minnesota Duluth)

  • Fang Wang

    (Capital Normal University)

  • Lin Zhang

    (University of Minnesota)

Abstract

Consider a p-variate normal random vector. We are interested in the limiting distributions of likelihood ratio test (LRT) statistics for testing the independence of its grouped components based on a random sample of size n. In classical multivariate analysis, the dimension p is fixed or relatively small, and the limiting distribution of the LRT is a chi-square distribution. When p goes to infinity, the chi-square approximation to the classical LRT statistic may be invalid. In this paper, we prove that the LRT statistic converges to a normal distribution under quite general conditions when p goes to infinity. We propose an adjusted test statistic which has a chi-square limit in general. Our comparison study indicates that the adjusted test statistic outperforms among the three approximations in terms of sizes. We also report some numerical results to compare the performance of our approaches and other methods in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:4:d:10.1007_s10463-018-0666-9
    DOI: 10.1007/s10463-018-0666-9
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. James R. Schott, 2005. "Testing for complete independence in high dimensions," Biometrika, Biometrika Trust, vol. 92(4), pages 951-956, December.
    5. Schott, James R., 2007. "A test for the equality of covariance matrices when the dimension is large relative to the sample sizes," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6535-6542, August.
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    Cited by:

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    3. 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.

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