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Lévy-frailty copulas

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  • Mai, Jan-Frederik
  • Scherer, Matthias

Abstract

A parametric family of n-dimensional extreme-value copulas of Marshall-Olkin type is introduced. Members of this class arise as survival copulas in Lévy-frailty models. The underlying probabilistic construction introduces dependence to initially independent exponential random variables by means of first-passage times of a Lévy subordinator. Jumps of the subordinator correspond to a singular component of the copula. Additionally, a characterization of completely monotone sequences via the introduced family of copulas is derived. An alternative characterization is given by Hausdorff's moment problem in terms of random variables with compact support. The resulting correspondence between random variables, Lévy subordinators, and copulas is studied and illustrated with several examples. Finally, it is used to provide a general methodology for sampling the copula in many cases. The new class is shown to share some properties with Archimedean copulas regarding construction and analytical form. Finally, the parametric form allows us to compute different measures of dependence and the Pickands representation.

Suggested Citation

  • Mai, Jan-Frederik & Scherer, Matthias, 2009. "Lévy-frailty copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1567-1585, August.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:7:p:1567-1585
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    References listed on IDEAS

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    1. Frahm, Gabriel, 2006. "On the extremal dependence coefficient of multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1470-1481, August.
    2. Friedrich Schmid & Rafael Schmidt, 2007. "Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(3), pages 323-354, November.
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    Citations

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    Cited by:

    1. Beer, Simone & Braun, Alexander & Marugg, Andrin, 2019. "Pricing industry loss warranties in a Lévy–Frailty framework," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 171-181.
    2. Paul Ressel, 2013. "Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences," Journal of Theoretical Probability, Springer, vol. 26(3), pages 666-675, September.
    3. Baglioni, Angelo & Cherubini, Umberto, 2013. "Within and between systemic country risk. Theory and evidence from the sovereign crisis in Europe," Journal of Economic Dynamics and Control, Elsevier, vol. 37(8), pages 1581-1597.
    4. Shenkman, Natalia, 2017. "A natural parametrization of multivariate distributions with limited memory," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 234-251.
    5. Brigo, Damiano & Mai, Jan-Frederik & Scherer, Matthias, 2016. "Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 60-66.
    6. Mai, Jan-Frederik & Scherer, Matthias, 2010. "The Pickands representation of survival Marshall-Olkin copulas," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 357-360, March.
    7. Fabrizio Durante & Marius Hofert & Matthias Scherer, 2010. "Multivariate Hierarchical Copulas with Shocks," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 681-694, December.
    8. Mai, Jan-Frederik, 2018. "Extreme-value copulas associated with the expected scaled maximum of independent random variables," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 50-61.
    9. Mai, Jan-Frederik & Scherer, Matthias, 2012. "H-extendible copulas," Journal of Multivariate Analysis, Elsevier, vol. 110(C), pages 151-160.
    10. Mai Jan-Frederik, 2014. "A note on the Galambos copula and its associated Bernstein function," Dependence Modeling, De Gruyter, vol. 2(1), pages 1-8, March.
    11. Ressel, Paul, 2011. "Monotonicity properties of multivariate distribution and survival functions -- With an application to Lévy-frailty copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 393-404, March.
    12. Durante, Fabrizio & Fernández Sánchez, Juan & Trutschnig, Wolfgang, 2014. "Multivariate copulas with hairpin support," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 323-334.
    13. Jan-Frederik Mai & Steffen Schenk & Matthias Scherer, 2017. "Two Novel Characterizations of Self-Decomposability on the Half-Line," Journal of Theoretical Probability, Springer, vol. 30(1), pages 365-383, March.
    14. Mai Jan-Frederik, 2020. "The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay," Dependence Modeling, De Gruyter, vol. 8(1), pages 210-220, January.
    15. Nadarajah, Saralees, 2015. "Expansions for bivariate copulas," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 77-84.
    16. Mai, Jan-Frederik & Scherer, Matthias & Shenkman, Natalia, 2013. "Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 457-480.

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