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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

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.

Suggested Citation

  • Stefan Aulbach & Verena Bayer & Michael Falk, 2012. "A multivariate piecing-together approach with an application to operational loss data," Papers 1205.1617, arXiv.org.
    • Handle: RePEc:arx:papers:1205.1617
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    1. 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.
    2. Marco Moscadelli, 2004. "The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee," Temi di discussione (Economic working papers) 517, Bank of Italy, Economic Research and International Relations Area.
    3. 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.
    4. Michel, René, 2008. "Some notes on multivariate generalized Pareto distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1288-1301, July.
    5. 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.
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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.
    2. Stefan Aulbach & Michael Falk & Timo Fuller, 2019. "Testing for a $$\delta $$ δ -neighborhood of a generalized Pareto copula," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 599-626, June.
    3. Falk, Michael & Padoan, Simone A. & Wisheckel, Florian, 2019. "Generalized Pareto copulas: A key to multivariate extremes," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    4. Falk, Michael & Stupfler, Gilles, 2017. "An offspring of multivariate extreme value theory: The max-characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 85-95.
    5. André, L.M. & Wadsworth, J.L. & O'Hagan, A., 2024. "Joint modelling of the body and tail of bivariate data," Computational Statistics & Data Analysis, Elsevier, vol. 189(C).
    6. Rootzen, Holger & Segers, Johan & Wadsworth, Jenny, 2016. "Multivariate peaks over thresholds models," LIDAM Discussion Papers ISBA 2016018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Aulbach, Stefan & Falk, Michael & Hofmann, Martin, 2012. "The multivariate Piecing-Together approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 161-170.

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