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Depth-based runs tests for bivariate central symmetry

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Abstract

McWilliams (J Am Stat Assoc 85:1130–1133, 1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the number of runs in the resulting series of signs is too small. This test is universally consistent and enjoys good robustness properties, but is unfortunately limited to the univariate setup. In this paper, we extend McWilliams’ procedure into tests of bivariate central symmetry. The proposed tests first reorder the observations according to their statistical depth in a symmetrized version of the sample, then reject the null when an original concept of simplicial runs is too small. Our tests are affine invariant and have good robustness properties. In particular, they do not require any finite moment assumption. We derive their limiting null distribution, which establishes their asymptotic distribution freeness. We study their finite-sample properties through Monte Carlo experiments and conclude with some final comments.

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Notes

  1. Zuo and Serfling (2000a) also consider P2 for weaker concepts of symmetry, namely angular and halfspace symmetry, but, for our purposes, we may restrict to central symmetry in the sequel.

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Acknowledgments

Christophe Ley thanks the Fonds National de la Recherche Scientifique, Communauté francaise de Belgique, for financial support via a Mandat d’Aspirant and a Mandat de Chargé de Recherche FNRS. The research of Davy Paindaveine was supported by an ARC grant of the Communauté Française de Belgique and by the IAP research network grant P7/06 of the Belgian government (Belgian Science Policy). The authors thank the two anonymous referees and the Editor for valuable comments that led to a significant improvement of the first version of this paper.

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Authors and Affiliations

  1. Seminar für Wirtschafts- und Sozialstatistik, Universität zu Köln, Meister-Ekkehart-Str. 9, 50923 , Cologne, Germany

    Rainer Dyckerhoff

  2. ECARES and Département de Mathématique, Université libre de Bruxelles, CP 210, Bld. du Triomphe, 1050 , Brussels, Belgium

    Christophe Ley

  3. ECARES and Département de Mathématique, Université libre de Bruxelles, CP 114/04, Av. F.D. Roosevelt, 50, 1050 , Brussels, Belgium

    Davy Paindaveine

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  1. Rainer Dyckerhoff
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  2. Christophe Ley
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Correspondence to Christophe Ley.

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Dyckerhoff, R., Ley, C. & Paindaveine, D. Depth-based runs tests for bivariate central symmetry. Ann Inst Stat Math 67, 917–941 (2015). https://doi.org/10.1007/s10463-014-0480-y

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  • DOI: https://doi.org/10.1007/s10463-014-0480-y

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