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A Two-Sample Test of High Dimensional Means Based on Posterior Bayes Factor

Author

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  • Yuanyuan Jiang

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China)

  • Xingzhong Xu

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
    Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China)

Abstract

In classical statistics, the primary test statistic is the likelihood ratio. However, for high dimensional data, the likelihood ratio test is no longer effective and sometimes does not work altogether. By replacing the maximum likelihood with the integral of the likelihood, the Bayes factor is obtained. The posterior Bayes factor is the ratio of the integrals of the likelihood function with respect to the posterior. In this paper, we investigate the performance of the posterior Bayes factor in high dimensional hypothesis testing through the problem of testing the equality of two multivariate normal mean vectors. The asymptotic normality of the linear function of the logarithm of the posterior Bayes factor is established. Then we construct a test with an asymptotically nominal significance level. The asymptotic power of the test is also derived. Simulation results and an application example are presented, which show good performance of the test. Hence, taking the posterior Bayes factor as a statistic in high dimensional hypothesis testing is a reasonable methodology.

Suggested Citation

  • Yuanyuan Jiang & Xingzhong Xu, 2022. "A Two-Sample Test of High Dimensional Means Based on Posterior Bayes Factor," Mathematics, MDPI, vol. 10(10), pages 1-23, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1741-:d:819173
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    References listed on IDEAS

    as
    1. Thulin, Måns, 2014. "A high-dimensional two-sample test for the mean using random subspaces," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 26-38.
    2. 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.
    3. Roger S. Zoh & Abhra Sarkar & Raymond J. Carroll & Bani K. Mallick, 2018. "A Powerful Bayesian Test for Equality of Means in High Dimensions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1733-1741, October.
    4. 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.
    5. T. Tony Cai & Weidong Liu & Yin Xia, 2014. "Two-sample test of high dimensional means under dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 349-372, March.
    6. Karl Bruce Gregory & Raymond J. Carroll & Veerabhadran Baladandayuthapani & Soumendra N. Lahiri, 2015. "A Two-Sample Test for Equality of Means in High Dimension," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 837-849, June.
    7. Zhang, Liang & Zhu, Tianming & Zhang, Jin-Ting, 2020. "A Simple Scale-Invariant Two-Sample Test for High-dimensional Data," Econometrics and Statistics, Elsevier, vol. 14(C), pages 131-144.
    8. Wang, Rui & Xu, Xingzhong, 2018. "On two-sample mean tests under spiked covariances," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 225-249.
    9. Jin-Ting Zhang & Jia Guo & Bu Zhou & Ming-Yen Cheng, 2020. "A Simple Two-Sample Test in High Dimensions Based on L2-Norm," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(530), pages 1011-1027, April.
    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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