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A triangle test for equality of distribution functions in high dimensions

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  • Zhenyu Liu
  • Reza Modarres

Abstract

A triangle statistic is proposed for testing the equality of two multivariate continuous distributions in high-dimensional settings based on sample interpoint distances. Given two independent p-dimensional random samples, a triangle can be formed by randomly selecting one observation from one sample and two observations from the other sample. The triangle statistic estimates the probability that the distance between the two observations from the same distribution is the largest, the middle or the smallest in the triangle being formed by these three observations. We show that the test based on the triangle statistic is asymptotically distribution-free under the null hypothesis of equal, but unknown continuous distribution functions. The triangle test is compared with other nonparametric tests through a simulation study. The triangle statistic is well defined when the number of variables p is larger than the number of observations m, and its computational complexity is independent of p, making it suitable for high-dimensional settings.

Suggested Citation

  • Zhenyu Liu & Reza Modarres, 2011. "A triangle test for equality of distribution functions in high dimensions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 605-615.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:3:p:605-615
    DOI: 10.1080/10485252.2010.485644
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    Citations

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    Cited by:

    1. Biswas, Munmun & Ghosh, Anil K., 2014. "A nonparametric two-sample test applicable to high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 160-171.
    2. Liu, Zhi & Xia, Xiaochao & Zhou, Wang, 2015. "A test for equality of two distributions via jackknife empirical likelihood and characteristic functions," Computational Statistics & Data Analysis, Elsevier, vol. 92(C), pages 97-114.
    3. Modarres, Reza, 2014. "On the interpoint distances of Bernoulli vectors," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 215-222.
    4. Reza Modarres & Yu Song, 2020. "Multivariate power series interpoint distances," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(4), pages 955-982, December.
    5. Lovato, Ilenia & Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2020. "Model-free two-sample test for network-valued data," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    6. 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).
    7. 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.
    8. Lingzhe Guo & Reza Modarres, 2020. "Testing the equality of matrix distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 289-307, June.
    9. Petrie, Adam, 2016. "Graph-theoretic multisample tests of equality in distribution for high dimensional data," Computational Statistics & Data Analysis, Elsevier, vol. 96(C), pages 145-158.
    10. Mondal, Pronoy K. & Biswas, Munmun & Ghosh, Anil K., 2015. "On high dimensional two-sample tests based on nearest neighbors," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 168-178.
    11. Reza Modarres, 2020. "Graphical Comparison of High‐Dimensional Distributions," International Statistical Review, International Statistical Institute, vol. 88(3), pages 698-714, December.

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