Construction of asymmetric multivariate copulas. (English) Zbl 1151.62043

Liebscher, Eckhard

J. Multivariate Anal. 99, No. 10, 2234-2250 (2008); erratum ibid. 102, No. 4, 869-870 (2011).

Summary: We introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
62H20	Measures of association (correlation, canonical correlation, etc.)

Keywords:

copula; Archimedean copula; tail dependence

References:

[1]	Alfonsi, A.; Brigo, D., New families of copulas based on periodic functions, Comm. statist. theory methods, 34, 1437-1447, (2005) · Zbl 1071.62047
[2]	Charpentier, A.; Segers, J., Lower tail dependence for Archimedean copulas: characterizations and pitfalls, Insur. math. econ., 40, 525-532, (2007) · Zbl 1183.62086
[3]	Chen, X.; Fan, Y.; Tsyrennikov, V., Efficient estimation of semiparametric multivariate copula models, J. amer. statist. assoc., 101, 1228-1240, (2006) · Zbl 1120.62312
[4]	Clayton, D.G., A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika, 65, 141-152, (1978) · Zbl 0394.92021
[5]	V. Durrleman, A. Nikeghbali, T. Roncalli, A simple transformation of copulas, Working Paper, Groupe de Recherche Opérationnelle Crédit Lyonnais, France, 2000
[6]	Frees, E.W.; Valdez, E.A., Understanding relationships using copulas, N. am. actuar. J., 2, 1, 1-5, (1998) · Zbl 1081.62564
[7]	Genest, C., Frank’s family of bivariate distributions, Biometrika, 74, 549-555, (1987) · Zbl 0635.62038
[8]	Genest, C.; Ghoudi, K.; Rivest, L.-P., A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika, 82, 543-552, (1995) · Zbl 0831.62030
[9]	Genest, C.; Ghoudi, K.; Rivest, L.-P., Discussion on the paper “understanding relationships using copulas” by E. frees and E. valdez, N. am. actuar. J., 2, 143-149, (1998)
[10]	Gumbel, E.J., Bivariate exponential distributions, J. amer. statist. assoc., 55, 698-707, (1960) · Zbl 0099.14501
[11]	Frank, M.J., On the simultaneous associativity of \(F(x, y)\) and \(x + y - F(x, y)\), Aequationes math., 19, 194-226, (1979) · Zbl 0444.39003
[12]	Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[13]	Joe, H., Asymptotic efficiency of the two-stage estimation method for copula-based models, J. multivariate anal., 94, 401-419, (2005) · Zbl 1066.62061
[14]	A. Khoudraji, Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées, Ph.D. Thesis, Université Laval Québec, Canada, 1995
[15]	Koehler, K.J.; Symanowski, J.T., Constructing multivariate distributions with specific marginal distributions, J. multivariate anal., 55, 261-282, (1995) · Zbl 0863.62048
[16]	Liebscher, E., Semiparametric density estimators using copulas, Comm. statist. theory methods, 34, 59-71, (2005) · Zbl 1066.62046
[17]	Marshall, A.W.; Olkin, I., Families of multivariate distributions, J. amer. statist. assoc., 83, 834-841, (1988) · Zbl 0683.62029
[18]	A.J. McNeil, J. Nešlehová, Multivariate Archimedean copulas, \(d\)-monotone functions and \(\ell_1\)-norm symmetric distributions. Ann. Statist. (2007) (in press)
[19]	Nelsen, R.B., Properties of a one-parameter family of bivariate distributions with specified marginals, Comm. stat. theory methods, 15, 3277-3285, (1986) · Zbl 0616.62067
[20]	Nelsen, R.B., ()