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On tests of radial symmetry for bivariate copulas

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  • Christian Genest
  • Johanna Nešlehová

Abstract

The unique copula of a continuous random pair $$(X,Y)$$ ( X , Y ) is said to be radially symmetric if and only if it is also the copula of the pair $$(-X,-Y)$$ ( - X , - Y ) . This paper revisits the recently considered issue of testing for radial symmetry. Three rank-based statistics are proposed to this end which are asymptotically equivalent but simpler to compute than those of Bouzebda and Cherfi (J Stat Plan Inference 142:1262–1271, 2012 ). Their limiting null distribution and its approximation using the multiplier bootstrap are discussed. The finite-sample properties of the resulting tests are assessed via simulations. The asymptotic distribution of one of the test statistics is also computed under an arbitrary alternative, thereby correcting an error in the recent work of Dehgani et al. (Stat Pap 54:271–286, 2013 ). Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Christian Genest & Johanna Nešlehová, 2014. "On tests of radial symmetry for bivariate copulas," Statistical Papers, Springer, vol. 55(4), pages 1107-1119, November.
    • Handle: RePEc:spr:stpapr:v:55:y:2014:i:4:p:1107-1119
    DOI: 10.1007/s00362-013-0556-4
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    References listed on IDEAS

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    1. Azam Dehgani & Ali Dolati & Manuel Úbeda-Flores, 2013. "Measures of radial asymmetry for bivariate random vectors," Statistical Papers, Springer, vol. 54(2), pages 271-286, May.
    2. Bücher, Axel & Dette, Holger, 2010. "A note on bootstrap approximations for the empirical copula process," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1925-1932, December.
    3. Rémillard, Bruno & Scaillet, Olivier, 2009. "Testing for equality between two copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 377-386, March.
    4. Neuhaus, Georg & Zhu, Li-Xing, 1998. "Permutation Tests for Reflected Symmetry," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 129-153, November.
    5. Segers, Johan, 2012. "Asymptotics of empirical copula processes under non-restrictive smoothness assumptions," LIDAM Reprints ISBA 2012009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Christian Genest & Johanna Nešlehová & Jean-François Quessy, 2012. "Tests of symmetry for bivariate copulas," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 811-834, August.
    7. J. Rosco & Harry Joe, 2013. "Measures of tail asymmetry for bivariate copulas," Statistical Papers, Springer, vol. 54(3), pages 709-726, August.
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    Cited by:

    1. Monica Billio & Lorenzo Frattarolo & Dominique Guégan, 2022. "High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization," Post-Print hal-04085236, HAL.
    2. Beare, Brendan K. & Seo, Juwon, 2020. "Randomization Tests Of Copula Symmetry," Econometric Theory, Cambridge University Press, vol. 36(6), pages 1025-1063, December.
    3. Shogo Kato & Toshinao Yoshiba & Shinto Eguchi, 2022. "Copula-based measures of asymmetry between the lower and upper tail probabilities," Statistical Papers, Springer, vol. 63(6), pages 1907-1929, December.
    4. Jean-François Quessy, 2021. "On nonparametric tests of multivariate meta-ellipticity," Statistical Papers, Springer, vol. 62(5), pages 2283-2310, October.
    5. Werner Hürlimann, 2017. "A comprehensive extension of the FGM copula," Statistical Papers, Springer, vol. 58(2), pages 373-392, June.
    6. Bahraoui Tarik & Bouezmarni Taoufik & Quessy Jean-François, 2018. "Testing the symmetry of a dependence structure with a characteristic function," Dependence Modeling, De Gruyter, vol. 6(1), pages 331-355, December.
    7. Juwon Seo, 2018. "Randomization Tests for Equality in Dependence Structure," Papers 1811.02105, arXiv.org.
    8. Miriam Jaser & Aleksey Min, 2021. "On tests for symmetry and radial symmetry of bivariate copulas towards testing for ellipticity," Computational Statistics, Springer, vol. 36(3), pages 1-26, September.
    9. Mangold, Benedikt, 2017. "New concepts of symmetry for copulas," FAU Discussion Papers in Economics 06/2017, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics, revised 2017.
    10. Billio Monica & Frattarolo Lorenzo & Guégan Dominique, 2021. "Multivariate radial symmetry of copula functions: finite sample comparison in the i.i.d case," Dependence Modeling, De Gruyter, vol. 9(1), pages 43-61, January.
    11. Pavel Krupskii, 2017. "Copula-based measures of reflection and permutation asymmetry and statistical tests," Statistical Papers, Springer, vol. 58(4), pages 1165-1187, December.
    12. Saminger-Platz Susanne & Kolesárová Anna & Šeliga Adam & Mesiar Radko & Klement Erich Peter, 2021. "New results on perturbation-based copulas," Dependence Modeling, De Gruyter, vol. 9(1), pages 347-373, January.
    13. Kojadinovic, Ivan, 2017. "Some copula inference procedures adapted to the presence of ties," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 24-41.
    14. Bücher Axel & Jaser Miriam & Min Aleksey, 2021. "Detecting departures from meta-ellipticity for multivariate stationary time series," Dependence Modeling, De Gruyter, vol. 9(1), pages 121-140, January.
    15. Quessy, Jean-François, 2021. "A Szekely–Rizzo inequality for testing general copula homogeneity hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    16. Axel Bücher & Ivan Kojadinovic, 2019. "A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1145-1165, September.
    17. Jean-François Quessy, 2019. "Consistent nonparametric tests for detecting gradual changes in the marginals and the copula of multivariate time series," Statistical Papers, Springer, vol. 60(3), pages 717-746, June.
    18. Jaser Miriam & Haug Stephan & Min Aleksey, 2017. "A simple non-parametric goodness-of-fit test for elliptical copulas," Dependence Modeling, De Gruyter, vol. 5(1), pages 330-353, December.

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