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Dependence Modeling

Ed. by Puccetti, Giovanni


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Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family

Kahadawala Cooray
Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/demo-2018-0001

Abstract

The family of Clayton copulas is one of the most widely used Archimedean copulas for dependency measurement. A major drawback of this copula is that when it accounts for negative dependence, the copula is nonstrict and its support is dependent on the parameter. The main motivation for this paper is to address this drawback by introducing a new two-parameter family of strict Archimedean copulas to measure exchangeable multivariate dependence. Closed-form formulas for both complete and d−monotonicity parameter regions of the generator and the copula distribution function are derived. In addition, recursive formulas for both copula and radial densities are obtained. Simulation studies are conducted to assess the performance of the maximum likelihood estimators of d−variate copula under known marginals. Furthermore, derivativefree closed-form formulas for Kendall’s distribution function are derived. A real multivariate data example is provided to illustrate the flexibility of the new copula for negative association.

Keywords: Archimedean copula; complete monotonicity; coverage probability; d−monotonicity; Kendall’s tau; Kendall’s distribution function

References

  • [1] Alsina, C., M. J. Frank, and B. Schweizer (2006). Associative Functions: Triangular Norms and Copulas. World Scientific Publishing, Hackensack NJ.Google Scholar

  • [2] Barbe, P., C. Genest, K. Ghoudi, and B. Rémillard (1996). On Kendall’s process. J. Multivariate Anal. 58(2), 197-229.CrossrefGoogle Scholar

  • [3] Capéraà, P. and C. Genest (1993). Spearman’s rho is larger than Kendall’s tau for positively dependent random variables. J. Nonparametr. Stat. 2(2), 183-194.CrossrefGoogle Scholar

  • [4] Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1), 141-151.Google Scholar

  • [5] Cook, R. D. and M. E. Johnson (1981). A family of distributions for modelling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. Ser. B 43(2), 210-218.Google Scholar

  • [6] Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.Google Scholar

  • [7] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL.Google Scholar

  • [8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Second edition. John Wiley & Sons, New York.Google Scholar

  • [9] Ferreira, C. and J. L. López (2004). Asymptotic expansions of the Hurwitz-Lerch Zeta function. J. Math. Anal. Appl. 298(1), 210-224.Google Scholar

  • [10] Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequationes Math. 19(1), 194-226.Google Scholar

  • [11] Fréchet, M. R. (1951). Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon, Sect. A (3) 14, 53-77.Google Scholar

  • [12] Fréchet, M. R. (1958). Remarques au sujet de la note précédente. C. R. Acad. Sci. Paris 246, 2719-2720.Google Scholar

  • [13] Genest, C., K. Ghoudi, and L.-P. Rivest (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543-552.CrossrefGoogle Scholar

  • [14] Genest, C. and J. MacKay (1986a). The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist. 40(4), 280-283.Google Scholar

  • [15] Genest, C. and J. MacKay (1986b). Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14(2), 145-159.Google Scholar

  • [16] Genest, C. and L.-P. Rivest (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88(423), 1034-1043.Google Scholar

  • [17] Genest, C., J. Nešlehová, and N. Ben Ghorbal (2011). Estimators based on Kendall’s tau in multivariate copula models. Aust. N. Z. J. Stat. 53(2), 157-177.Web of ScienceGoogle Scholar

  • [18] Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal. 52(12), 5163-5174.Google Scholar

  • [19] Hofert, M. (2011). E_cienctly sampling nested Archimedean copulas. Comput. Statist. Data Anal. 55(1), 57-70.Google Scholar

  • [20] Hofert, M., M. Mächler, and A. J. McNeil (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. J. Multivariate Anal. 110, 133-150.Web of ScienceGoogle Scholar

  • [21] Hoeffding, W. (1994). Scale-invariant correlation theory. In N. I. Fisher and P. K. Sen (Eds.), The Collected Works of Wassily Hoeffding, pp. 57-107. Springer, New York.Google Scholar

  • [22] Hoeffding,W. (1994). Scale-invariant correlationmeasures for discontinuous distributions. In N. I. Fisher and P. K. Sen (Eds.), The Collected Works of Wassily Hoe_ding, pp. 109-133. Springer, New York.Google Scholar

  • [23] Joe, H. (1990). Multivariate concordance. J. Multivariate Anal. 35(1), 12-30.Google Scholar

  • [24] Joe, H. (1993). Parametric families of multivariate distributions with given margins. J. Multivariate Anal. 46(2), 262-282.Google Scholar

  • [25] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar

  • [26] Kimeldorf, G. and A. R. Sampson (1975). Uniform representations of bivariate distributions. Comm. Statist. 4(7), 617-627.Google Scholar

  • [27] Lehmann, E. L. and G. Casella (1998). Theory of Point Estimation. Second edition. Springer, New York.Google Scholar

  • [28] Liebscher, E. (2008). Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99(10), 2234-2250.Web of ScienceGoogle Scholar

  • [29] McNeil, A. J. and J. Nešlehová (2009). Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions. Ann. Statist. 37(5B), 3059-3097.Google Scholar

  • [30] McNeil, A. J. and J. Nešlehová (2010). From Archimedean to Liouville copulas. J. Multivariate Anal. 101(8), 1772-1790.Google Scholar

  • [31] Nelsen, R. B. (1997). Dependence and order in families of Archimedean copulas. J. Multivariate Anal. 60(1), 111-122.Google Scholar

  • [32] Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Google Scholar

  • [33] Srivastava, H. M. and J. Choi (2001). Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht.Google Scholar

  • [34] Widder, D. V. (1941). The Laplace Transform. Princeton University Press, Princeton NJ.Google Scholar

About the article

Received: 2017-02-25

Accepted: 2017-12-28

Published Online: 2018-02-07


Citation Information: Dependence Modeling, Volume 6, Issue 1, Pages 1–18, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2018-0001.

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© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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