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A multivariate piecing-together approach with an application to operational loss data

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  • Stefan Aulbach
  • Verena Bayer
  • Michael Falk
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    Abstract

    The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuous manner. The result is again a copula. The other step consists of the transformation of each margin of this new copula by a given univariate distribution function. This provides, altogether, a multivariate distribution function with prescribed margins whose copula coincides in its central part with $C$ and in its upper tail with a GPD copula. When applied to data, this approach also enables the evaluation of a wide range of rational scenarios for the upper tail of the underlying distribution function in the multivariate case. We apply this approach to operational loss data in order to evaluate the range of operational risk.

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    File URL: http://arxiv.org/pdf/1205.1617
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1205.1617.

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    Date of creation: May 2012
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    Publication status: Published in Bernoulli 2012, Vol. 18, No. 2, 455-475
    Handle: RePEc:arx:papers:1205.1617

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    Web page: http://arxiv.org/

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    1. 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.
    2. Genest, C. & Quesada Molina, J. J. & Rodriguez Lallena, J. A. & Sempi, C., 1999. "A Characterization of Quasi-copulas," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 193-205, May.
    3. Michel, René, 2008. "Some notes on multivariate generalized Pareto distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1288-1301, July.
    4. Alsina, Claudi & Nelsen, Roger B. & Schweizer, Berthold, 1993. "On the characterization of a class of binary operations on distribution functions," Statistics & Probability Letters, Elsevier, vol. 17(2), pages 85-89, May.
    5. repec:sae:ecolab:v:16:y:2006:i:2:p:1-2 is not listed on IDEAS
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    Cited by:
    1. Durante, Fabrizio & Fernández Sánchez, Juan & Sempi, Carlo, 2013. "Multivariate patchwork copulas: A unified approach with applications to partial comonotonicity," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 897-905.

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