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Higher order tail densities of copulas and hidden regular variation

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  • Li, Haijun
  • Hua, Lei

Abstract

A notion of higher order tail densities for copulas is introduced using multivariate regular variation of copula densities, and densities of multivariate extremes with various margins can then be studied in a unified fashion. We show that the tail of a multivariate density can be decomposed into the tail density of the underlying copula, coupled with marginal tail transforms of the three types: Fréchet, Gumbel, and Weibull types. We also derive the relation between the tail density and tail order functions of a copula in the context of hidden regular variation. Some illustrative examples are given.

Suggested Citation

  • Li, Haijun & Hua, Lei, 2015. "Higher order tail densities of copulas and hidden regular variation," Journal of Multivariate Analysis, Elsevier, vol. 138(C), pages 143-155.
    • Handle: RePEc:eee:jmvana:v:138:y:2015:i:c:p:143-155
    DOI: 10.1016/j.jmva.2014.12.010
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    References listed on IDEAS

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    1. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    2. Joe, Harry & Li, Haijun & Nikoloulopoulos, Aristidis K., 2010. "Tail dependence functions and vine copulas," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 252-270, January.
    3. Hua, Lei & Joe, Harry, 2012. "Tail comonotonicity: Properties, constructions, and asymptotic additivity of risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 492-503.
    4. de Haan, L. & Omey, E., 1984. "Integrals and derivatives of regularly varying functions in d and domains of attraction of stable distributions II," Stochastic Processes and their Applications, Elsevier, vol. 16(2), pages 157-170, February.
    5. Hua, Lei & Joe, Harry, 2012. "Tail Comonotonicity and Conservative Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 42(2), pages 601-629, November.
    6. Harry Joe & Haijun Li, 2011. "Tail Risk of Multivariate Regular Variation," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 671-693, December.
    7. de Haan, L. & Resnick, S., 1987. "On regular variation of probability densities," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 83-93.
    8. Hua, Lei, 2015. "Tail negative dependence and its applications for aggregate loss modeling," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 135-145.
    9. Li, Haijun, 2009. "Orthant tail dependence of multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 243-256, January.
    10. Hua, Lei & Joe, Harry, 2011. "Tail order and intermediate tail dependence of multivariate copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1454-1471, November.
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    Citations

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    Cited by:

    1. Das Bikramjit & Fasen-Hartmann Vicky, 2019. "Conditional excess risk measures and multivariate regular variation," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 1-23, December.
    2. Haijun Li, 2018. "Operator Tail Dependence of Copulas," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1013-1027, September.
    3. Joe, Harry & Li, Haijun, 2019. "Tail densities of skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 421-435.
    4. Das, Bikramjit & Fasen-Hartmann, Vicky, 2024. "On heavy-tailed risks under Gaussian copula: The effects of marginal transformation," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    5. Takaaki Koike & Marius Hofert, 2020. "Modality for Scenario Analysis and Maximum Likelihood Allocation," Papers 2005.02950, arXiv.org, revised Nov 2020.
    6. Lei Hua, 2016. "A Note on Upper Tail Behavior of Liouville Copulas," Risks, MDPI, vol. 4(4), pages 1-10, November.
    7. Hua, Lei, 2017. "On a bivariate copula with both upper and lower full-range tail dependence," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 94-104.

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