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The multivariate Piecing-Together approach revisited

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  • Aulbach, Stefan
  • Falk, Michael
  • Hofmann, Martin

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. A multivariate extension was established by Aulbach et al. (in press) [2]: the upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (in press) [2], we achieve in the present paper an exact representation of the PT-copula’s upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.

Suggested Citation

  • Aulbach, Stefan & Falk, Michael & Hofmann, Martin, 2012. "The multivariate Piecing-Together approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 161-170.
    • Handle: RePEc:eee:jmvana:v:110:y:2012:i:c:p:161-170
    DOI: 10.1016/j.jmva.2012.02.002
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    References listed on IDEAS

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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. 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. Stefan Aulbach & Verena Bayer & Michael Falk, 2012. "A multivariate piecing-together approach with an application to operational loss data," Papers 1205.1617, arXiv.org.
    4. Charpentier, Arthur & Segers, Johan, 2009. "Tails of multivariate Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1521-1537, August.
    5. Segers, Johan, 2012. "Asymptotics of empirical copula processes under non-restrictive smoothness assumptions," LIDAM Reprints ISBA 2012009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Michel, René, 2008. "Some notes on multivariate generalized Pareto distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1288-1301, July.
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    Cited by:

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    3. 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.
    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.

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