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Permutation tests for equality of distributions in high-dimensional settings

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  • Peter Hall

Abstract

Motivated by applications in high-dimensional settings, we suggest a test of the hypothesis H-sub-0 that two sampled distributions are identical. It is assumed that two independent datasets are drawn from the respective populations, which may be very general. In particular, the distributions may be multivariate or infinite-dimensional, in the latter case representing, for example, the distributions of random functions from one Euclidean space to another. Our test uses a measure of distance between data. This measure should be symmetric but need not satisfy the triangle inequality, so it is not essential that it be a metric. The test is based on ranking the pooled dataset, with respect to the distance and relative to any fixed data value, and repeating this operation for each fixed datum. A permutation argument enables a critical point to be chosen such that the test has concisely known significance level, conditional on the set of all pairwise distances. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • Peter Hall, 2002. "Permutation tests for equality of distributions in high-dimensional settings," Biometrika, Biometrika Trust, vol. 89(2), pages 359-374, June.
    • Handle: RePEc:oup:biomet:v:89:y:2002:i:2:p:359-374
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    Cited by:

    1. Jun Li & Jifei Ban & Louis S. Santiago, 2011. "Nonparametric Tests for Homogeneity of Species Assemblages: A Data Depth Approach," Biometrics, The International Biometric Society, vol. 67(4), pages 1481-1488, December.
    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. Federico A. Bugni & Joel L. Horowitz, 2021. "Permutation tests for equality of distributions of functional data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(7), pages 861-877, November.
    4. Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2018. "Hotelling’s T2 in separable Hilbert spaces," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 284-305.
    5. 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.
    6. 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.
    7. 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).
    8. 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).
    9. Qiu, Zhiping & Chen, Jianwei & Zhang, Jin-Ting, 2021. "Two-sample tests for multivariate functional data with applications," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    10. A. Pini & S. Vantini, 2017. "Interval-wise testing for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 407-424, April.
    11. 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.
    12. W. Lok & Stephen Lee, 2011. "A new statistical depth function with applications to multimodal data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 617-631.
    13. Liu, Jiamin & Ma, Shuangge & Xu, Wangli & Zhu, Liping, 2022. "A generalized Wilcoxon–Mann–Whitney type test for multivariate data through pairwise distance," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    14. Duncan Simester & Artem Timoshenko & Spyros I. Zoumpoulis, 2020. "Targeting Prospective Customers: Robustness of Machine-Learning Methods to Typical Data Challenges," Management Science, INFORMS, vol. 66(6), pages 2495-2522, June.
    15. Ruiyi Zhang & R. Todd Ogden & Martin Picard & Anuj Srivastava, 2022. "Nonparametric k‐sample test on shape spaces with applications to mitochondrial shape analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(1), pages 51-69, January.
    16. 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.
    17. Pini, Alessia & Spreafico, Lorenzo & Vantini, Simone & Vietti, Alessandro, 2019. "Multi-aspect local inference for functional data: Analysis of ultrasound tongue profiles," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 162-185.
    18. Reza Modarres, 2020. "Graphical Comparison of High‐Dimensional Distributions," International Statistical Review, International Statistical Institute, vol. 88(3), pages 698-714, December.
    19. Qiu, Tao & Zhang, Qintong & Fang, Yuanyuan & Xu, Wangli, 2024. "Testing homogeneity in high dimensional data through random projections," Journal of Multivariate Analysis, Elsevier, vol. 200(C).
    20. Marc Ditzhaus & Daniel Gaigall, 2022. "Testing marginal homogeneity in Hilbert spaces with applications to stock market returns," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 749-770, September.

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