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Linear B-spline copulas with applications to nonparametric estimation of copulas

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  • Shen, Xiaojing
  • Zhu, Yunmin
  • Song, Lixin

Abstract

In this paper, we propose a method for constructing a new class of copulas. They are called linear B-spline copulas which are a good approximation of a given complicated copula by using finite numbers of values of this copula without the loss of some essential properties. Moreover, rigorous analysis shows that the empirical linear B-spline copulas are more effective than empirical copulas to estimate perfectly dependent copulas. For the cases of nonperfectly dependent copulas, simulations show that the empirical linear B-spline copulas also improve the empirical copulas to estimate the underlying copula structure. Furthermore, we compare the performance of parametric estimation of copulas based on the empirical copulas with that based on the empirical linear B-spline copulas by simulations. In most of the cases, the latter are better than the former.

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  • 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.
    • Handle: RePEc:eee:csdana:v:52:y:2008:i:7:p:3806-3819
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    References listed on IDEAS

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    1. Fermanian, Jean-David & Scaillet, Olivier, 2003. "Nonparametric estimation of copulas for time series," Working Papers unige:41797, University of Geneva, Geneva School of Economics and Management.
    2. Dhaene, Jan & Goovaerts, Marc J., 1996. "Dependency of Risks and Stop-Loss Order1," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 201-212, November.
    3. Kim, Gunky & Silvapulle, Mervyn J. & Silvapulle, Paramsothy, 2007. "Comparison of semiparametric and parametric methods for estimating copulas," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 2836-2850, March.
    4. Michel Denuit, 2004. "Nonparametric Tests for Positive Quadrant Dependence," Journal of Financial Econometrics, Oxford University Press, vol. 2(3), pages 422-450.
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    Cited by:

    1. 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.
    2. Dalla Valle, Luciana & De Giuli, Maria Elena & Tarantola, Claudia & Manelli, Claudio, 2016. "Default probability estimation via pair copula constructions," European Journal of Operational Research, Elsevier, vol. 249(1), pages 298-311.
    3. Romera, Rosario & Molanes, Elisa M., 2008. "Copulas in finance and insurance," DES - Working Papers. Statistics and Econometrics. WS ws086321, Universidad Carlos III de Madrid. Departamento de Estadística.
    4. Diks, Cees & Panchenko, Valentyn & van Dijk, Dick, 2010. "Out-of-sample comparison of copula specifications in multivariate density forecasts," Journal of Economic Dynamics and Control, Elsevier, vol. 34(9), pages 1596-1609, September.
    5. Eddie Anderson & Artem Prokhorov & Yajing Zhu, 2020. "A Simple Estimator of Two‐Dimensional Copulas, with Applications," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(6), pages 1375-1412, December.
    6. Marra, Giampiero & Radice, Rosalba, 2017. "Bivariate copula additive models for location, scale and shape," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 99-113.
    7. 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.
    8. 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.
    9. Nadja Klein & Thomas Kneib & Giampiero Marra & Rosalba Radice & Slawa Rokicki & Mark E. McGovern, 2018. "Mixed Binary-Continuous Copula Regression Models with Application to Adverse Birth Outcomes," CHaRMS Working Papers 18-06, Centre for HeAlth Research at the Management School (CHaRMS).
    10. Hernández-Lobato, José Miguel & Suárez, Alberto, 2011. "Semiparametric bivariate Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2038-2058, June.
    11. Giampiero Marra & Rosalba Radice & Till Bärnighausen & Simon N. Wood & Mark E. McGovern, 2017. "A Simultaneous Equation Approach to Estimating HIV Prevalence With Nonignorable Missing Responses," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 484-496, April.
    12. Xiaoling Dou & Satoshi Kuriki & Gwo Dong Lin & Donald Richards, 2021. "Dependence Properties of B-Spline Copulas," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 283-311, February.
    13. Gregor Wei{ss} & Marcus Scheffer, 2012. "Smooth Nonparametric Bernstein Vine Copulas," Papers 1210.2043, arXiv.org.

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