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From Archimedean to Liouville copulas

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  • McNeil, Alexander J.
  • Neslehová, Johanna
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    Abstract

    We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Neslehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall's tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall's tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.

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    Bibliographic Info

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

    Volume (Year): 101 (2010)
    Issue (Month): 8 (September)
    Pages: 1772-1790

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    Handle: RePEc:eee:jmvana:v:101:y:2010:i:8:p:1772-1790

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    Keywords: Archimedean copula Simplex distribution l1-norm symmetric distribution Liouville distribution Kendall's tau Williamson d-transform Laplace transform Stochastic ordering Dependence ordering Stochastic simulation;

    References

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    1. Hofert, Marius, 2008. "Sampling Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5163-5174, August.
    2. Fang, Kai-Tai & Fang, Bi-Qi, 1988. "Some families of mutivariate symmetric distributions related to exponential distribution," Journal of Multivariate Analysis, Elsevier, vol. 24(1), pages 109-122, January.
    3. Gupta, Rameshwar D. & Richards, Donald St. P., 1992. "Multivariate Liouville distributions, III," Journal of Multivariate Analysis, Elsevier, vol. 43(1), pages 29-57, October.
    4. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
    5. Gupta, Rameshwar D. & Richards, Donald St.P., 1987. "Multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 233-256, December.
    6. Joe, Harry, 1993. "Multivariate dependence measures and data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 16(3), pages 279-297, September.
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    Citations

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    Cited by:
    1. Paul Embrechts & Marius Hofert, 2011. "Comments on: Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(2), pages 263-270, August.
    2. Philippe Lambert, 2011. "Comments on: Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(2), pages 284-286, August.
    3. Christian Genest & Johanna Nešlehová & Jean-François Quessy, 2012. "Tests of symmetry for bivariate copulas," Annals of the Institute of Statistical Mathematics, Springer, vol. 64(4), pages 811-834, August.
    4. Hua, Lei & Joe, Harry, 2014. "Strength of tail dependence based on conditional tail expectation," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 143-159.
    5. Charpentier, A. & Fougères, A.-L. & Genest, C. & Nešlehová, J.G., 2014. "Multivariate Archimax copulas," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 118-136.
    6. Paul Janssen & Luc Duchateau, 2011. "Comments on: Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(2), pages 271-275, August.
    7. El Alaoui, Marwane & Benbachir, Saâd, 2012. "Spillover Effect in the MENA Area: Case of Four Financial Markets," MPRA Paper 48682, University Library of Munich, Germany.
    8. Christian Genest & Johanna Nešlehová & Johanna Ziegel, 2011. "Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(2), pages 223-256, August.

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