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Distribution-free high-dimensional two-sample tests based on discriminating hyperplanes

Author

Listed:
  • Anil K. Ghosh

    (Indian Statistical Institute)

  • Munmun Biswas

    (Indian Statistical Institute)

Abstract

In this article, we propose a general procedure for multivariate generalizations of univariate distribution-free tests involving two independent samples as well as matched pair data. This proposed procedure is based on ranks of real-valued linear functions of multivariate observations. The linear function used to rank the observations is obtained by solving a classification problem between the two multivariate distributions from which the observations are generated. Our proposed tests retain the distribution-free property of their univariate analogs, and they perform well for high-dimensional data even when the dimension exceeds the sample size. Asymptotic results on their power properties are derived when the dimension grows to infinity and the sample size may or may not grow with the dimension. We analyze several high-dimensional simulated and real data sets to compare the empirical performance of our proposed tests with several other tests available in the literature.

Suggested Citation

  • Anil K. Ghosh & Munmun Biswas, 2016. "Distribution-free high-dimensional two-sample tests based on discriminating hyperplanes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 525-547, September.
    • Handle: RePEc:spr:testjl:v:25:y:2016:i:3:d:10.1007_s11749-015-0467-x
    DOI: 10.1007/s11749-015-0467-x
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    References listed on IDEAS

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    1. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444, June.
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    6. Rousson, Valentin, 2002. "On Distribution-Free Tests for the Multivariate Two-Sample Location-Scale Model," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 43-57, January.
    7. Qiao, Xingye & Zhang, Hao Helen & Liu, Yufeng & Todd, Michael J. & Marron, J. S., 2010. "Weighted Distance Weighted Discrimination and Its Asymptotic Properties," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 401-414.
    8. J. Cuesta-Albertos & M. Febrero-Bande, 2010. "A simple multiway ANOVA for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 537-557, November.
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    Cited by:

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    2. Jiang, Qing & Hušková, Marie & Meintanis, Simos G. & Zhu, Lixing, 2019. "Asymptotics, finite-sample comparisons and applications for two-sample tests with functional data," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 202-220.
    3. Paul, Biplab & De, Shyamal K. & Ghosh, Anil K., 2022. "Some clustering-based exact distribution-free k-sample tests applicable to high dimension, low sample size data," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    4. Shin-ichi Tsukada, 2019. "High dimensional two-sample test based on the inter-point distance," Computational Statistics, Springer, vol. 34(2), pages 599-615, June.
    5. 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).

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