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Dependence and Order in Families of Archimedean Copulas

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  • Nelsen, Roger B.
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    Abstract

    The copula for a bivariate distribution functionH(x, y) with marginal distribution functionsF(x) andG(y) is the functionCdefined byH(x, y)=C(F(x), G(y)).Cis called Archimedean ifC(u, v)=[phi]-1([phi](u)+[phi](v)), where[phi]is a convex decreasing continuous function on (0, 1] with[phi](1)=0. A copula has lower tail dependence ifC(u, u)/uconverges to a constant[gamma]in (0, 1] asu-->0+; and has upper tail dependence ifC(u, u)/(1-u) converges to a constant[delta]in (0, 1] asu-->1-whereCdenotes the survival function corresponding toC. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values of[gamma]and[delta], and present extensions to higher dimensions. We also investigate limiting cases and the concordance ordering of these families. In the process, we present answers to two open problems posed by Joe (1993,J. Multivariate Anal.46262-282).

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 60 (1997)
    Issue (Month): 1 (January)
    Pages: 111-122

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    Handle: RePEc:eee:jmvana:v:60:y:1997:i:1:p:111-122

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    Related research

    Keywords: Archimedean copula bivariate distribution multivariate distribution concordance ordering lower tail dependence upper tail dependence;

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    Cited by:
    1. Ostap Okhrin & Yarema Okhrin & Wolfgang Schmid, 2009. "Properties of Hierarchical Archimedean Copulas," SFB 649 Discussion Papers SFB649DP2009-014, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    2. A. Sancetta & Satchell, S.E., 2001. "Bernstein Approximations to the Copula Function and Portfolio Optimization," Cambridge Working Papers in Economics 0105, Faculty of Economics, University of Cambridge.
    3. Huang, Jen-Jsung & Lee, Kuo-Jung & Liang, Hueimei & Lin, Wei-Fu, 2009. "Estimating value at risk of portfolio by conditional copula-GARCH method," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 315-324, December.
    4. Frahm, Gabriel, 2006. "On the extremal dependence coefficient of multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1470-1481, August.
    5. Mulero, Julio & Pellerey, Franco & Rodríguez-Griñolo, Rosario, 2010. "Stochastic comparisons for time transformed exponential models," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 328-333, April.
    6. Mesfioui, Mhamed & Quessy, Jean-François, 2008. "Dependence structure of conditional Archimedean copulas," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 372-385, March.

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