MULTIVARIATE ARCHIMEDEAN COPULAS, d-MONOTONE FUNCTIONS AND l_i-NORM SYMMETRIC DISTRIBUTIONS

摘要

It is shown that a necessary and sufficient condition for an Archimede_an copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copu_las is shown to coincide with the class of survival copulas of d-dimensional i1-norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189_207] in an analogous manner to the well-known Bernstein_Widder characteri_zation of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Ar_chimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of sin_gular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.