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On optimal tests for rotational symmetry against new classes of hyperspherical distributions

Author

Listed:
  • Eduardo Garcia-Portugues

    (Carlos III University of Madrid)

  • Davy Paindaveine

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Thomas Verdebout

    (ECARES - European Center for Advanced Research in Economics and Statistics - ULB - Université libre de Bruxelles)

Abstract

Motivated by the central role played by rotationally symmetric distributions in directionalstatistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopta semiparametric approach and tackle problems where the location of the symmetry axis iseither specified or unspecified. For each problem, we define two tests and study their asymptoticproperties under very mild conditions. We introduce two new classes of directional distributionsthat extend the rotationally symmetric class and are of independent interest. We prove thateach test is locally asymptotically maximin, in the Le Cam sense, for one kind of the alternativesgiven by the new classes of distributions, both for specified and unspecified symmetry axis. Thetests, aimed to detect location-like and scatter-like alternatives, are combined into convenienthybrid tests that are consistent against both alternatives. We perform Monte Carlo experimentsthat illustrate the finite-sample performances of the proposed tests and their agreement withthe asymptotic results. Finally, the practical relevance of our tests is illustrated on a real dataapplication from astronomy. The R packagerotasymimplements the proposed tests and allowspractitioners to reproduce the data application

Suggested Citation

  • Eduardo Garcia-Portugues & Davy Paindaveine & Thomas Verdebout, 2020. "On optimal tests for rotational symmetry against new classes of hyperspherical distributions," Post-Print hal-03169388, HAL.
    • Handle: RePEc:hal:journl:hal-03169388
    DOI: 10.1080/01621459.2019.1665527
    Note: View the original document on HAL open archive server: https://hal.science/hal-03169388
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    References listed on IDEAS

    as
    1. Ley, Christophe & Verdebout, Thomas, 2017. "Skew-rotationally-symmetric distributions and related efficient inferential procedures," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 67-81.
    2. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Testing Uniformity on High-Dimensional Spheres against Contiguous Rotationally Symmetric Alternatives," Working Papers ECARES ECARES 2015-04, ULB -- Universite Libre de Bruxelles.
    3. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Xu, Danli & Wang, Yong, 2023. "Density estimation for spherical data using nonparametric mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).
    2. Davy Paindaveine & Joséa Rasoafaraniaina & Thomas Verdebout, 2021. "Preliminary test estimation in uniformly locally asymptotically normal models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 689-707, June.
    3. Janice L. Scealy, 2021. "Comments on: 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 68-70, March.
    4. Dabo-Niang, Sophie & Thiam, Baba & Verdebout, Thomas, 2022. "Asymptotic efficiency of some nonparametric tests for location on hyperspheres," Statistics & Probability Letters, Elsevier, vol. 188(C).
    5. Kim, Byungwon & Schulz, Jörn & Jung, Sungkyu, 2020. "Kurtosis test of modality for rotationally symmetric distributions on hyperspheres," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    6. Marc Hallin & H Lui & Thomas Verdebout, 2022. "Nonparametric Measure-transportation-based Methods for Directional Data," Working Papers ECARES 2022-18, ULB -- Universite Libre de Bruxelles.

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    More about this item

    Keywords

    Directional data; Hypothesis testing; Local asymptotic normality; Locally asymptotically maximin tests; Rotational symmetry;
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