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Probit Transformation for Nonparametric Kernel Estimation of the Copula Density


  • Gery Geenens
  • Arthur Charpentier
  • Davy Paindaveine


Copula modelling has become ubiquitous in modern statistics. Here, the problem of nonparametricallyestimating a copula density is addressed. Arguably the most popular nonparametric density estimator,the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it isheavily a↵ected by boundary bias issues. In addition, most common copulas admit unbounded densities,and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimatoris proposed. It is based on the idea of transforming the uniform marginals of the copula density intonormal distributions via the probit function, estimating the density in the transformed domain, whichcan be accomplished without boundary problems, and obtaining an estimate of the copula densitythrough back-transformation. Although natural, a raw application of this procedure was, however, seennot to perform very well in the earlier literature. Here, it is shown that, if combined with local likelihooddensity estimation methods, the idea yields very good and easy to implement estimators, fixing boundaryissues in a natural way and able to cope with unbounded copula densities. The asymptotic properties ofthe suggested estimators are derived, and a practical way of selecting the crucially important smoothingparameters is devised. Finally, extensive simulation studies and a real data analysis evidence theirexcellent performance compared to their main competitors.

Suggested Citation

  • Gery Geenens & Arthur Charpentier & Davy Paindaveine, 2014. "Probit Transformation for Nonparametric Kernel Estimation of the Copula Density," Working Papers ECARES ECARES 2014-23, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/159977

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    References listed on IDEAS

    1. Faugeras, Olivier P., 2009. "A quantile-copula approach to conditional density estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2083-2099, October.
    2. Genest, Christian & Rémillard, Bruno & Beaudoin, David, 2009. "Goodness-of-fit tests for copulas: A review and a power study," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 199-213, April.
    3. Chen, Xiaohong & Fan, Yanqin & Pouzo, Demian & Ying, Zhiliang, 2010. "Estimation and model selection of semiparametric multivariate survival functions under general censorship," Journal of Econometrics, Elsevier, vol. 157(1), pages 129-142, July.
    4. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1837-1845, September.
    5. Göran Kauermann & Christian Schellhase & David Ruppert, 2013. "Flexible Copula Density Estimation with Penalized Hierarchical B-splines," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 685-705, December.
    6. Jean-David FERMANIAN & Olivier SCAILLET, 2003. "Nonparametric Estimation of Copulas for Time Series," FAME Research Paper Series rp57, International Center for Financial Asset Management and Engineering.
    7. Duong, Tarn & Hazelton, Martin L., 2005. "Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 417-433, April.
    8. Genest, Christian & Masiello, Esterina & Tribouley, Karine, 2009. "Estimating copula densities through wavelets," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 170-181, April.
    9. Bouezmarni, Taoufik & Rombouts, Jeroen V.K. & Taamouti, Abderrahim, 2010. "Asymptotic properties of the Bernstein density copula estimator for [alpha]-mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 1-10, January.
    10. Autin, F. & Le Pennec, E. & Tribouley, K., 2010. "Thresholding methods to estimate copula density," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 200-222, January.
    11. Christian Genest & Bruno Rémillard, 2004. "Test of independence and randomness based on the empirical copula process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 335-369, December.
    12. Bo Li & Marc G. Genton, 2013. "Nonparametric Identification of Copula Structures," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(502), pages 666-675, June.
    13. Wolfgang Härdle & Ostap Okhrin, 2009. "De copulis non est disputandum - Copulae: An Overview," SFB 649 Discussion Papers SFB649DP2009-031, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    14. Einmahl, J. H. J. & Ruymgaart, F. H., 1987. "The almost sure behavior of the oscillation modulus of the multivariate empirical process," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 87-96, November.
    15. Mack, Y. P. & Rosenblatt, M., 1979. "Multivariate k-nearest neighbor density estimates," Journal of Multivariate Analysis, Elsevier, vol. 9(1), pages 1-15, March.
    16. Klugman, Stuart A. & Parsa, Rahul, 1999. "Fitting bivariate loss distributions with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 139-148, March.
    17. repec:bpj:strimo:v:3:y:1985:i:3-4:p:239-262:n:4 is not listed on IDEAS
    18. Fermanian, Jean-David, 2005. "Goodness-of-fit tests for copulas," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 119-152, July.
    19. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    20. Bücher, Axel & Volgushev, Stanislav, 2013. "Empirical and sequential empirical copula processes under serial dependence," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 61-70.
    21. Shen, Xiaojing & Zhu, Yunmin & Song, Lixin, 2008. "Linear B-spline copulas with applications to nonparametric estimation of copulas," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3806-3819, March.
    22. Scaillet, Olivier, 2007. "Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 533-543, March.
    23. Qu, Leming & Yin, Wotao, 2012. "Copula density estimation by total variation penalized likelihood with linear equality constraints," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 384-398.
    24. Gery Geenens, 2014. "Probit Transformation for Kernel Density Estimation on the Unit Interval," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 346-358, March.
    25. Paul Embrechts, 2009. "Copulas: A Personal View," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 76(3), pages 639-650.
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    Cited by:

    1. Gery Geenens & Richard Dunn, 2017. "A nonparametric copula approach to conditional Value-at-Risk," Papers 1712.05527,
    2. Nagler, Thomas & Czado, Claudia, 2016. "Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 69-89.
    3. Rodrigues, G.S. & Nott, David J. & Sisson, S.A., 2016. "Functional regression approximate Bayesian computation for Gaussian process density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 229-241.
    4. Otneim, Håkon & Tjøstheim, Dag, 2016. "Non-parametric estimation of conditional densities: A new method," Discussion Papers 2016/22, Norwegian School of Economics, Department of Business and Management Science.

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    copula density; transformation kernel density estimator; boundary bias; unbounded Density; local likelihood density estimation;

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