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Depth-based runs tests for bivariate central symmetry

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  • Rainer Dyckerhoff
  • Christophe Ley
  • Davy Paindaveine

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. Copyright The Institute of Statistical Mathematics, Tokyo 2015

Suggested Citation

  • Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2015. "Depth-based runs tests for bivariate central symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(5), pages 917-941, October.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:5:p:917-941
    DOI: 10.1007/s10463-014-0480-y
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    References listed on IDEAS

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    1. Neuhaus, Georg & Zhu, Li-Xing, 1998. "Permutation Tests for Reflected Symmetry," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 129-153, November.
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    3. Davy Paindaveine & Germain Van bever, 2013. "From Depth to Local Depth: A Focus on Centrality," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(503), pages 1105-1119, September.
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    7. Henze, N. & Klar, B. & Meintanis, S. G., 2003. "Invariant tests for symmetry about an unspecified point based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 275-297, November.
    8. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
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    Cited by:

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    2. Morgunov, V.I. (Моргунов, В.И.), 2016. "The Liquidity Management of the Banking Sector and the Short-Term Money Market Interest Rates [Управление Ликвидностью Банковского Сектора И Краткосрочной Процентной Ставкой Денежного Рынка]," Working Papers 21311, Russian Presidential Academy of National Economy and Public Administration.
    3. Sladana Babic & Laetitia Gelbgras & Marc Hallin & Christophe Ley, 2019. "Optimal tests for elliptical symmetry: specified and unspecified location," Working Papers ECARES 2019-26, ULB -- Universite Libre de Bruxelles.
    4. Ricardo Fraiman & Leonardo Moreno & Sebastian Vallejo, 2017. "Some hypothesis tests based on random projection," Computational Statistics, Springer, vol. 32(3), pages 1165-1189, September.
    5. Ivanović, Blagoje & Milošević, Bojana & Obradović, Marko, 2020. "Comparison of symmetry tests against some skew-symmetric alternatives in i.i.d. and non-i.i.d. setting," Computational Statistics & Data Analysis, Elsevier, vol. 151(C).
    6. Sakineh Dehghan & Mohammad Reza Faridrohani, 2019. "Affine invariant depth-based tests for the multivariate one-sample location problem," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 671-693, September.
    7. Sakineh Dehghan & Mohammad Reza Faridrohani & Zahra Barzegar, 2023. "Testing for diagonal symmetry based on center-outward ranking," Statistical Papers, Springer, vol. 64(1), pages 255-283, February.
    8. Van Bever, Germain, 2016. "Simplicial bivariate tests for randomness," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 20-25.

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