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On certain transformation of Archimedean copulas: Application to the non-parametric estimation of their generators

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  • Elena Di Bernardino

    (CEDRIC - Centre d'études et de recherche en informatique et communications - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - CNAM - Conservatoire National des Arts et Métiers [CNAM] - HESAM - HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université)

  • Didier Rullière

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the $r$-fold composition of the diagonal section of a copula, from $r \in \N$ to $r \in \R$. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these expressions are proposed and their convergence is investigated. We provide confidence bands for the estimated generators. Numerical illustrations show the empirical performance of these estimators.

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  • Elena Di Bernardino & Didier Rullière, 2013. "On certain transformation of Archimedean copulas: Application to the non-parametric estimation of their generators," Post-Print hal-00834000, HAL.
    • Handle: RePEc:hal:journl:hal-00834000
    DOI: 10.2478/demo-2013-0001
    Note: View the original document on HAL open archive server: https://hal.science/hal-00834000v3
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    Cited by:

    1. Elena Di Bernardino & Didier Rullière, 2017. "A note on upper-patched generators for Archimedean copulas," Post-Print hal-01347869, HAL.
    2. Elena Di Bernardino & Didier Rullière, 2016. "A note on upper-patched generators for Archimedean copulas," Working Papers hal-01347869, HAL.
    3. Fabrizio Durante & Ostap Okhrin, 2014. "Estimation procedures for exchangeable Marshall copulas with hydrological application," SFB 649 Discussion Papers SFB649DP2014-014, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Elena Di Bernardino & Didier Rullière, 2016. "On tail dependence coefficients of transformed multivariate Archimedean copulas," Post-Print hal-00992707, HAL.
    5. Di Bernardino Elena & Rullière Didier, 2016. "On an asymmetric extension of multivariate Archimedean copulas based on quadratic form," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-20, December.
    6. Elena Di Bernardino & Didier Rullière, 2016. "On an asymmetric extension of multivariate Archimedean copulas based on quadratic form," Working Papers hal-01147778, HAL.
    7. Durante Fabrizio & Fernández-Sánchez Juan & Trutschnig Wolfgang, 2014. "Solution to an open problem about a transformation on the space of copulas," Dependence Modeling, De Gruyter, vol. 2(1), pages 1-8, November.
    8. Fadal A.A. Aldhufairi & Jungsywan H. Sepanski, 2020. "New families of bivariate copulas via unit weibull distortion," Journal of Statistical Distributions and Applications, Springer, vol. 7(1), pages 1-20, December.
    9. repec:bpj:demode:v:6:y:2018:i:1:p:139-155:n:9 is not listed on IDEAS
    10. Elena Di Bernardino & Didier Rullière, 2015. "Estimation of multivariate critical layers: Applications to rainfall data," Post-Print hal-00940089, HAL.
    11. Fadal Abdullah-A Aldhufairi & Ranadeera G.M. Samanthi & Jungsywan H. Sepanski, 2020. "New Families of Bivariate Copulas via Unit Lomax Distortion," Risks, MDPI, vol. 8(4), pages 1-19, October.
    12. Elena Di Bernardino & Clémentine Prieur, 2014. "Estimation of multivariate conditional-tail-expectation using Kendall's process," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(2), pages 241-267, June.
    13. Girard Stéphane, 2018. "Transformation Of A Copula Using The Associated Co-Copula," Dependence Modeling, De Gruyter, vol. 6(1), pages 298-308, December.

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    Keywords

    Transformations of Archimedean copula; self-nested diagonal; non-parametric estimation; tail dependence;
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