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Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances

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  • Ebner, Bruno
  • Henze, Norbert
  • Yukich, Joseph E.

Abstract

We present a unified approach to goodness-of-fit testing in Rd and on lower-dimensional manifolds embedded in Rd based on sums of powers of weighted volumes of kth nearest neighbor spheres. We prove asymptotic normality of a class of test statistics under the null hypothesis and under fixed alternatives. Under such alternatives, scaled versions of the test statistics converge to the α-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests. The tests are applied to two data sets of gamma-ray bursts in astronomy.

Suggested Citation

  • Ebner, Bruno & Henze, Norbert & Yukich, Joseph E., 2018. "Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 231-242.
    • Handle: RePEc:eee:jmvana:v:165:y:2018:i:c:p:231-242
    DOI: 10.1016/j.jmva.2017.12.009
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    References listed on IDEAS

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    1. 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.
    2. Dette, H. & Henze, N., 1990. "Some peculiar boundary phenomena for extremes of rth nearest neighbor links," Statistics & Probability Letters, Elsevier, vol. 10(5), pages 381-390, October.
    3. Jupp, P. E., 2001. "Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 1-20, April.
    4. Biau, Gérard & Devroye, Luc & Dujmović, Vida & Krzyżak, Adam, 2012. "An affine invariant k-nearest neighbor regression estimate," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 24-34.
    5. Petrie, Adam & Willemain, Thomas R., 2013. "An empirical study of tests for uniformity in multidimensional data," Computational Statistics & Data Analysis, Elsevier, vol. 64(C), pages 253-268.
    6. M. Coleman Miller, 2017. "A golden binary," Nature, Nature, vol. 551(7678), pages 36-37, November.
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    Cited by:

    1. Aya-Moreno, Carlos & Geenens, Gery & Penev, Spiridon, 2018. "Shape-preserving wavelet-based multivariate density estimation," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 30-47.
    2. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    3. Bruno Ebner & Norbert Henze & Simos Meintanis, 2024. "A unified approach to goodness-of-fit testing for spherical and hyperspherical data," Statistical Papers, Springer, vol. 65(6), pages 3447-3475, August.
    4. Solveig Flaig & Gero Junike, 2023. "Validation of machine learning based scenario generators," Papers 2301.12719, arXiv.org, revised Nov 2023.
    5. Solveig Flaig & Gero Junike, 2021. "Scenario generation for market risk models using generative neural networks," Papers 2109.10072, arXiv.org, revised Aug 2023.

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