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On testing the equality of high dimensional mean vectors with unequal covariance matrices

Author

Listed:
  • Jiang Hu

    (Northeast Normal University)

  • Zhidong Bai

    (Northeast Normal University)

  • Chen Wang

    (National University of Singapore)

  • Wei Wang

    (Northeast Normal University)

Abstract

In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problems in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by Bai and Saranadasa (Stat Sin 6:311–329, 1996) and Chen and Qin (Ann Stat 38:808–835, 2010), we introduce a test statistic and derive the asymptotic distributions under the null and the alternative hypothesis. In addition, it is compared with a test statistic recently proposed by Srivastava and Kubokawa (J Multivar Anal 115:204–216, 2013). It is shown that our test statistic performs better especially in the large dimensional case.

Suggested Citation

  • Jiang Hu & Zhidong Bai & Chen Wang & Wei Wang, 2017. "On testing the equality of high dimensional mean vectors with unequal covariance matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 365-387, April.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:2:d:10.1007_s10463-015-0543-8
    DOI: 10.1007/s10463-015-0543-8
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    References listed on IDEAS

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    1. Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    2. Srivastava, Muni S. & Fujikoshi, Yasunori, 2006. "Multivariate analysis of variance with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 1927-1940, October.
    3. Cai, T. Tony & Xia, Yin, 2014. "High-dimensional sparse MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 174-196.
    4. Srivastava, Muni S., 2009. "A test for the mean vector with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 518-532, March.
    5. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    6. Srivastava, Muni S. & Yanagihara, Hirokazu, 2010. "Testing the equality of several covariance matrices with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1319-1329, July.
    7. Srivastava, Muni S. & Kubokawa, Tatsuya, 2013. "Tests for multivariate analysis of variance in high dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 204-216.
    8. Srivastava, Muni S. & Kollo, Tõnu & von Rosen, Dietrich, 2011. "Some tests for the covariance matrix with fewer observations than the dimension under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1090-1103, July.
    9. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    10. Srivastava, Muni S. & Du, Meng, 2008. "A test for the mean vector with fewer observations than the dimension," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 386-402, March.
    11. Srivastava, Muni S. & Katayama, Shota & Kano, Yutaka, 2013. "A two sample test in high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 349-358.
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    Cited by:

    1. Bai, Yansong & Zhang, Yong & Liu, Congmin, 2023. "Moderate deviation principle for likelihood ratio test in multivariate linear regression model," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    2. Zhang, Jin-Ting & Zhou, Bu & Guo, Jia, 2022. "Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference L2-norm based test," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    3. Jamshid Namdari & Debashis Paul & Lili Wang, 2021. "High-Dimensional Linear Models: A Random Matrix Perspective," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 645-695, August.
    4. Lixiu Wu & Jiang Hu, 2024. "Multi-sample hypothesis testing of high-dimensional mean vectors under covariance heterogeneity," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(4), pages 579-615, August.
    5. Zhou, Bu & Guo, Jia, 2017. "A note on the unbiased estimator of Σ2," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 141-146.
    6. Dörnemann, Nina, 2023. "Likelihood ratio tests under model misspecification in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    7. Zhang, Qiuyan & Wang, Chen & Zhang, Baoxue & Yang, Hu, 2024. "An RIHT statistic for testing the equality of several high-dimensional mean vectors under homoskedasticity," Computational Statistics & Data Analysis, Elsevier, vol. 190(C).
    8. Cousido-Rocha, Marta & de Uña-Álvarez, Jacobo & Hart, Jeffrey D., 2019. "A two-sample test for the equality of univariate marginal distributions for high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    9. Hyodo, Masashi & Watanabe, Hiroki & Seo, Takashi, 2018. "On simultaneous confidence interval estimation for the difference of paired mean vectors in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 160-173.
    10. M. Rauf Ahmad, 2019. "A unified approach to testing mean vectors with large dimensions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(4), pages 593-618, December.
    11. Chen, Jiaqi & Zhang, Yangchun & Li, Weiming & Tian, Boping, 2018. "A supplement on CLT for LSS under a large dimensional generalized spiked covariance model," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 57-65.
    12. Zhidong Bai & Jiang Hu & Chen Wang & Chao Zhang, 2021. "Test on the linear combinations of covariance matrices in high-dimensional data," Statistical Papers, Springer, vol. 62(2), pages 701-719, April.
    13. Harrar, Solomon W. & Kong, Xiaoli, 2022. "Recent developments in high-dimensional inference for multivariate data: Parametric, semiparametric and nonparametric approaches," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    14. Jin-Ting Zhang & Bu Zhou & Jia Guo, 2022. "Testing high-dimensional mean vector with applications," Statistical Papers, Springer, vol. 63(4), pages 1105-1137, August.
    15. Ping‐Shou Zhong & Jun Li & Piotr Kokoszka, 2021. "Multivariate analysis of variance and change points estimation for high‐dimensional longitudinal data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 375-405, June.

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