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Asymptotic properties of the Bernstein density copula estimator for alpha-mixing data

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  • BOUEZMARNI, Taoufik
  • ROMBOUTS, Jeroen VK
  • TAAMOUTI, Abderrahim

Abstract

Copulas are extensively used for dependence modeling. In many cases the data does not reveal how the dependence can be modeled using a particular parametric copula. Nonparametric copulas do not share this problem since they are entirely data based. This paper proposes nonparametric estimation of the density copula for [alpha]-mixing data using Bernstein polynomials. We focus only on the dependence structure between stochastic processes, captured by the copula density defined on the unit cube, and not the complete distribution. We study the asymptotic properties of the Bernstein density copula, i.e., we provide the exact asymptotic bias and variance, we establish the uniform strong consistency and the asymptotic normality. An empirical application is considered to illustrate the dependence structure among international stock markets (US and Canada) using the Bernstein density copula estimator.
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  • BOUEZMARNI, Taoufik & ROMBOUTS, Jeroen VK & TAAMOUTI, Abderrahim, 2010. "Asymptotic properties of the Bernstein density copula estimator for alpha-mixing data," LIDAM Reprints CORE 2302, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:2302
    DOI: 10.1016/j.jmva.2009.02.014
    Note: In : Journal of Statistical Planning and Inference, 140(1), 139-152, 2010
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    7. Sonia Petrone, 1999. "Random Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 373-393, September.
    8. Bouezmarni, T. & Rombouts, J.V.K., 2009. "Semiparametric multivariate density estimation for positive data using copulas," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2040-2054, April.
    9. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    10. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(3), pages 535-562, June.
    11. Axel Tenbusch, 1997. "Nonparametric curve estimation with bernstein estimates," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 1-30, January.
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    Cited by:

    1. Noh, Hohsuk & El Ghouch, Anouar & Bouezmarni, Taoufik, 2012. "Copula-Based Regression Estimation and Inference," LIDAM Discussion Papers ISBA 2012010, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Bouezmarni, Taoufik & El Ghouch, Anouar & Taamouti, Abderrahim, 2011. "Bernstein estimator for unbounded density copula," UC3M Working papers. Economics we1143, Universidad Carlos III de Madrid. Departamento de Economía.
    3. Berghaus, Betina & Segers, Johan, 2018. "Weak convergence of the weighted empirical beta copula process," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 266-281.
    4. Mirza Nazmul Hasan & Roel Braekers, 2022. "Modelling the association in bivariate survival data by using a Bernstein copula," Computational Statistics, Springer, vol. 37(2), pages 781-815, April.
    5. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    6. Taamouti, Abderrahim & Bouezmarni, Taoufik & El Ghouch, Anouar, 2014. "Nonparametric estimation and inference for conditional density based Granger causality measures," Journal of Econometrics, Elsevier, vol. 180(2), pages 251-264.
    7. 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.
    8. repec:cte:werepe:we1212 is not listed on IDEAS
    9. Taoufik Bouezmarni & Jeroen V.K. Rombouts & Abderrahim Taamouti, 2011. "Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(2), pages 275-287, October.
    10. 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.
    11. Taoufik Bouezmarni & Mohamed Doukali & Abderrahim Taamouti, 2023. "Copula-based estimation of health inequality measures with an application to COVID-19," University of East Anglia School of Economics Working Paper Series 2023-01, School of Economics, University of East Anglia, Norwich, UK..
    12. Bouezmarni Taoufik & Ghouch El & Taamouti Abderrahim, 2013. "Bernstein estimator for unbounded copula densities," Statistics & Risk Modeling, De Gruyter, vol. 30(4), pages 343-360, December.
    13. Taoufik Bouezmarni & Mohamed Doukali & Abderrahim Taamouti, 2022. "Copula-based estimation of health concentration curves with an application to COVID-19," CIRANO Working Papers 2022s-07, CIRANO.
    14. Taamouti, Abderrahim & Bouezmarni, Taoufik & El Ghouch, Anouar, 2012. "Nonparametric estimation and inference for Granger causality measures," UC3M Working papers. Economics 14150, Universidad Carlos III de Madrid. Departamento de Economía.
    15. Janssen, Paul & Swanepoel, Jan & Veraverbeke, Noël, 2014. "A note on the asymptotic behavior of the Bernstein estimator of the copula density," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 480-487.
    16. Okhrin Ostap, 2013. "Editorial to the special issue on Copulae of Statistics & Risk Modeling," Statistics & Risk Modeling, De Gruyter, vol. 30(4), pages 281-286, December.

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