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On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions

Author

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  • Eduardo García-Portugués
  • Davy Paindaveine
  • Thomas Verdebout

Abstract

Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt a semiparametric approach and tackle problems where the location of the symmetry axis is either specified or unspecified. For each problem, we define two tests and study their asymptotic properties under very mild conditions. We introduce two new classes of directional distributions that extend the rotationally symmetric class and are of independent interest. We prove that each test is locally asymptotically maximin, in the Le Cam sense, for one kind of the alternatives given by the new classes of distributions, for both specified and unspecified symmetry axis. The tests, aimed to detect location- and scatter-like alternatives, are combined into convenient hybrid tests that are consistent against both alternatives. We perform Monte Carlo experiments that illustrate the finite-sample performances of the proposed tests and their agreement with the asymptotic results. Finally, the practical relevance of our tests is illustrated on a real data application from astronomy. The R package rotasym implements the proposed tests and allows practitioners to reproduce the data application. Supplementary materials for this article are available online.

Suggested Citation

  • Eduardo García-Portugués & Davy Paindaveine & Thomas Verdebout, 2020. "On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(532), pages 1873-1887, December.
    • Handle: RePEc:taf:jnlasa:v:115:y:2020:i:532:p:1873-1887
    DOI: 10.1080/01621459.2019.1665527
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    Citations

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    Cited by:

    1. Maxime Boucher & Andrea Meilán-Vila & Vivien Meurice & Thomas Verdebout, 2025. "On a modified Watson test for spherical location," Statistical Papers, Springer, vol. 66(4), pages 1-12, June.
    2. Anahita Nodehi & Mousa Golalizadeh & Mehdi Maadooliat & Claudio Agostinelli, 2025. "Torus Probabilistic Principal Component Analysis," Journal of Classification, Springer;The Classification Society, vol. 42(2), pages 435-456, July.
    3. María Alonso‐Pena & Gerda Claeskens & Irène Gijbels, 2024. "Nonparametric estimation of densities on the hypersphere using a parametric guide," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(3), pages 956-986, September.
    4. Yu, Zehao & Huang, Xianzheng, 2025. "Regression analysis of elliptically symmetric directional data," Computational Statistics & Data Analysis, Elsevier, vol. 208(C).
    5. Alberto Fernández-de-Marcos & Eduardo García-Portugués, 2023. "On new omnibus tests of uniformity on the hypersphere," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(4), pages 1508-1529, December.
    6. 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).
    7. 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.
    8. Xu, Danli & Wang, Yong, 2023. "Density estimation for spherical data using nonparametric mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).
    9. Jeong Min Jeon & Ingrid Van Keilegom, 2024. "Density estimation and regression analysis on hyperspheres in the presence of measurement error," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 513-556, June.
    10. Ludwig Baringhaus & Rudolf Grübel, 2024. "Discrete mixture representations of spherical distributions," Statistical Papers, Springer, vol. 65(2), pages 557-596, April.
    11. 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.
    12. 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).
    13. Bernard, Gaspard & Verdebout, Thomas, 2024. "On some multivariate sign tests for scatter matrix eigenvalues," Econometrics and Statistics, Elsevier, vol. 29(C), pages 252-260.
    14. Fernández-de-Marcos, Alberto & García-Portugués, Eduardo, 2024. "A stereographic test of spherical uniformity," Statistics & Probability Letters, Elsevier, vol. 215(C).
    15. 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.
    16. 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.

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