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A Note on Upper Tail Behavior of Liouville Copulas


  • Lei Hua

    () (Division of Statistics, Northern Illinois University, DeKalb, IL 60115, USA)


The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson’s d -transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the upper tails, and they can be useful in modeling tail risks. In this article, we study the upper tail behavior of Liouville copulas through their upper tail orders. Tail orders of a more general scale mixture model that covers Liouville distributions is first derived, and then tail order functions and tail order density functions of Liouville copulas are derived. Concrete examples are given after the main results.

Suggested Citation

  • Lei Hua, 2016. "A Note on Upper Tail Behavior of Liouville Copulas," Risks, MDPI, Open Access Journal, vol. 4(4), pages 1-10, November.
  • Handle: RePEc:gam:jrisks:v:4:y:2016:i:4:p:40-:d:82313

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    References listed on IDEAS

    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. Alexandra Ramos & Anthony Ledford, 2009. "A new class of models for bivariate joint tails," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 219-241.
    4. 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.
    5. Saralees Nadarajah & Samuel Kotz*, 2005. "On the Product and Ratio of Gamma and Beta Random Variables," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 89(4), pages 435-449, November.
    6. 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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    1. repec:eee:jmvana:v:160:y:2017:i:c:p:68-92 is not listed on IDEAS

    More about this item


    tail order; tail order functions; tail order density; dirichlet distributions; scale mixtures;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law


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