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Estimation of multivariate critical layers: Applications to rainfall data


Author Info

  • Elena Di Bernardino

    (CEDRIC - Centre d'Etude et De Recherche en Informatique du Cnam - Conservatoire National des Arts et Métiers (CNAM))

  • Didier Rullière

    (SAF - Laboratoire de Sciences Actuarielle et Financière - Université Claude Bernard - Lyon I (UCBL) : EA2429)


Calculating return periods and critical layers (i.e., multivariate quantile curves) in a multivariate environment is a di cult problem. A possible consistent theoretical framework for the calculation of the return period, in a multi-dimensional environment, is essentially based on the notion of copula and level sets of the multivariate probability distribution. In this paper we propose a fast and parametric methodology to estimate the multivariate critical layers of a distribution and its associated return periods. The model is based on transformations of the marginal distributions and transformations of the dependence structure within the class of Archimedean copulas. The model has a tunable number of parameters, and we show that it is possible to get a competitive estimation without any global optimum research. We also get parametric expressions for the critical layers and return periods. The methodology is illustrated on hydrological 5-dimensional real data. On this real data-set we obtain a good quality of estimation and we compare the obtained results with some classical parametric competitors

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Bibliographic Info

Paper provided by HAL in its series Working Papers with number hal-00940089.

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Date of creation: 02 Jun 2014
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Handle: RePEc:hal:wpaper:hal-00940089

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Keywords: Multivariate probability transformations; level sets estimation; copulas; hyperbolic conversion functions; risk assessment; multivariate return periods.;

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  1. Alexis Bienven�e & Didier Rulli�re, 2012. "Iterative Adjustment of Survival Functions by Composed Probability Distortions," The Geneva Risk and Insurance Review, Palgrave Macmillan, vol. 37(2), pages 156-179, September.
  2. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, Elsevier, vol. 53(4), pages 391-399, July.
  3. Paul Embrechts & Marius Hofert, 2011. "Comments on: Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, Springer, vol. 20(2), pages 263-270, August.
  4. Segers, Johan & Uyttendaele, Nathan, 2014. "Nonparametric estimation of the tree structure of a nested Archimedean copula," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 72(C), pages 190-204.
  5. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 97(2), pages 526-547, February.
  6. Di Bernardino Elena & Rullière Didier, 2013. "On certain transformations of Archimedean copulas: Application to the non-parametric estimation of their generators," Dependence Modeling, De Gruyter Open, De Gruyter Open, vol. 1, pages 1-36, October.
  7. Barbe, Philippe & Genest, Christian & Ghoudi, Kilani & Rémillard, Bruno, 1996. "On Kendall's Process," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 58(2), pages 197-229, August.
  8. 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.
  9. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Penev, Spiridon I., 2008. "GeD spline estimation of multivariate Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 52(7), pages 3570-3582, March.
  10. Ivan Kojadinovic & Jun Yan, . "Modeling Multivariate Distributions with Continuous Margins Using the copula R Package," Journal of Statistical Software, American Statistical Association, American Statistical Association, vol. 34(i09).
  11. Hofert, Marius & Pham, David, 2013. "Densities of nested Archimedean copulas," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 118(C), pages 37-52.
  12. Nelsen, Roger B. & Quesada-Molina, José Juan & Rodri­guez-Lallena, José Antonio & Úbeda-Flores, Manuel, 2008. "On the construction of copulas and quasi-copulas with given diagonal sections," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 473-483, April.
  13. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 51(10), pages 5112-5129, June.
  14. Di Bernardino, Elena & Rullière, Didier, 2013. "Distortions of multivariate distribution functions and associated level curves: Applications in multivariate risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 190-205.
  15. Wysocki, Włodzimierz, 2012. "Constructing archimedean copulas from diagonal sections," Statistics & Probability Letters, Elsevier, Elsevier, vol. 82(4), pages 818-826.
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Cited by:
  1. Elena Di Bernardino & Didier Rullière, 2014. "On tail dependence coefficients of transformed multivariate Archimedean copulas," Working Papers hal-00992707, HAL.


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