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A Simple Scale-Invariant Two-Sample Test for High-dimensional Data

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  • Zhang, Liang
  • Zhu, Tianming
  • Zhang, Jin-Ting

Abstract

A new scale-invariant test for two-sample problems for high-dimensional data is proposed and studied. Under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-square-type mixture are shown to have the same limiting distribution after they are normalized. The limiting distribution can be normal or non-normal, depending on the underlying covariance structure of the high-dimensional data. To approximate the null distribution of the proposed test, the well-known Welch-Satterthwaite chi-square approximation is applied. The resulting test is shown to be adaptive to the shape of the underlying null distribution in the sense that when the test statistic is asymptotically normally distributed under the null hypothesis, so is the approximation distribution, and when the approximation distribution is asymptotically non-normally distributed, so is the underlying null distribution of the test statistic. The asymptotic powers of the proposed test under some local alternatives are derived. Simulation studies and a real data application are used to demonstrate the good performance of the proposed test compared with several existing competitors in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ecosta:v:14:y:2020:i:c:p:131-144
    DOI: 10.1016/j.ecosta.2019.12.002
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    References listed on IDEAS

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    1. Yujun Wu & Marc G. Genton & Leonard A. Stefanski, 2006. "A Multivariate Two-Sample Mean Test for Small Sample Size and Missing Data," Biometrics, The International Biometric Society, vol. 62(3), pages 877-885, September.
    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. 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.
    4. Jin-Ting Zhang, 2005. "Approximate and Asymptotic Distributions of Chi-Squared-Type Mixtures With Applications," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 273-285, March.
    5. 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.
    6. 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. 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.

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