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Distortions of multivariate risk measures: a level-sets based approach

Author

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  • Elena Di Bernardino

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

  • Didier Rullière

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

Abstract

In this paper, we propose a parametric model for multivariate distributions. The model is based on distortion functions, i.e. some transformations of a multivariate distribution which permit to generate new families of multivariate distribution functions. We derive some properties of considered distortions. A suitable proximity indicator between level curves is introduced in order to evaluate the quality of candidate distortion parameters. Using this proximity indicator and properties of distorted level curves, we give a specific estimation procedure. The estimation algorithm is mainly relying on straightforward univariate optimizations, and we finally get parametric representations of both multivariate distribution functions and associated level curves. Our results are motivated by applications in multivariate risk theory. The methodology is illustrated on real examples.

Suggested Citation

  • Elena Di Bernardino & Didier Rullière, 2012. "Distortions of multivariate risk measures: a level-sets based approach," Working Papers hal-00756387, HAL.
  • Handle: RePEc:hal:wpaper:hal-00756387 Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00756387
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    References listed on IDEAS

    as
    1. Alexis Bienvenüe & Didier Rullière, 2012. "Iterative Adjustment of Survival Functions by Composed Probability Distortions," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 37(2), pages 156-179, September.
    2. Biernacki, Christophe & Celeux, Gilles & Govaert, Gerard, 2003. "Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 561-575, January.
    3. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    4. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    5. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    6. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
    7. Chacón, José E. & Rodríguez-Casal, Alberto, 2010. "A note on the universal consistency of the kernel distribution function estimator," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1414-1419, September.
    8. Dehaan, L. & Huang, X., 1995. "Large Quantile Estimation in a Multivariate Setting," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 247-263, May.
    9. Valdez, Emiliano A., 2009. "On the Distortion of a Copula and its Margins," MPRA Paper 20524, University Library of Munich, Germany.
    10. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
    11. Alexis Bienvenüe & Didier Rullière, 2011. "On hyperbolic iterated distortions for the adjustment of survival functions," Post-Print hal-00665349, HAL.
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