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Higher order tail densities of copulas and hidden regular variation

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Listed:
  • 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. Hua, Lei, 2015. "Tail negative dependence and its applications for aggregate loss modeling," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 135-145.
    3. 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.
    4. 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.
    5. 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.
    6. Hua, Lei & Joe, Harry, 2012. "Tail Comonotonicity and Conservative Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 42(2), pages 601-629, November.
    7. 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.
    8. de Haan, L. & Resnick, S., 1987. "On regular variation of probability densities," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 83-93.
    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. Joe, Harry & Li, Haijun, 2019. "Tail densities of skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 421-435.
    2. Takaaki Koike & Marius Hofert, 2020. "Modality for Scenario Analysis and Maximum Likelihood Allocation," Papers 2005.02950, arXiv.org, revised Nov 2020.
    3. 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.
    4. Lei Hua, 2016. "A Note on Upper Tail Behavior of Liouville Copulas," Risks, MDPI, vol. 4(4), pages 1-10, November.
    5. 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.
    6. Haijun Li, 2018. "Operator Tail Dependence of Copulas," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 1013-1027, September.

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