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Nonparametric Measure-transportation-based Methods for Directional Data

Author

Listed:
  • Marc Hallin
  • H Lui
  • Thomas Verdebout

Abstract

This paper proposes various nonparametric tools for directional data based on measure transportation. We use optimal transports todefine new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closedformformulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They yield fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper) meridians. Based on this, we also propose a test of uniformity and establish its universal consistency; simulations indicate that this test outperforms the “projected” Cram´er-von Mises, Anderson-Darling, and Rothman procedures recently proposed in the literature. Two real-data examples involving the analysis of Venus craters and proteins structures conclude the paper.

Suggested Citation

  • 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.
    • Handle: RePEc:eca:wpaper:2013/344268
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    References listed on IDEAS

    as
    1. Christophe Ley & Yvik Swan & Thomas Verdebout, 2017. "Efficient ANOVA for directional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 39-62, February.
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    Keywords

    directional statistics; quantile; ranks; optimal transport;
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