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Graphical Comparison of High‐Dimensional Distributions

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  • Reza Modarres

Abstract

We consider groups of observations in Rd and present a simultaneous plot of the empirical cumulative distribution functions of the within and between interpoint distances to visualise and examine the equality of the underlying distribution functions of the observations. We provide several examples to illustrate how such plots can be utilised to envision and canvass the relationship between the two distributions under location, scale, dependence and shape changes. We suggest new statistics for testing the equality of k distributions and extend the simultaneous plots to visualise them. We compare the new statistics to existing tests based on the interpoint distances.

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  • Reza Modarres, 2020. "Graphical Comparison of High‐Dimensional Distributions," International Statistical Review, International Statistical Institute, vol. 88(3), pages 698-714, December.
    • Handle: RePEc:bla:istatr:v:88:y:2020:i:3:p:698-714
    DOI: 10.1111/insr.12358
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    References listed on IDEAS

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    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. Paul R. Rosenbaum, 2005. "An exact distribution‐free test comparing two multivariate distributions based on adjacency," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 515-530, September.
    3. Yu Song & Reza Modarres, 2019. "Interpoint Distance Test of Homogeneity for Multivariate Mixture Models," International Statistical Review, International Statistical Institute, vol. 87(3), pages 613-638, December.
    4. 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.
    5. D. Zhan & J. D. Hart, 2014. "Testing equality of a large number of densities," Biometrika, Biometrika Trust, vol. 101(2), pages 449-464.
    6. Giovagnoli, Alessandra & Wynn, H. P., 1995. "Multivariate dispersion orderings," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 325-332, March.
    7. Peter Hall, 2002. "Permutation tests for equality of distributions in high-dimensional settings," Biometrika, Biometrika Trust, vol. 89(2), pages 359-374, June.
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    Cited by:

    1. Modarres, Reza, 2022. "A high dimensional dissimilarity measure," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    2. Modarres, Reza, 2023. "Analysis of distance matrices," Statistics & Probability Letters, Elsevier, vol. 193(C).
    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).

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