Multivariate Archimedean copulas, \(d\)-monotone functions and \(\ell _{1}\)-norm symmetric distributions. (English) Zbl 1173.62044
Summary: It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a \(d\)-dimensional copula is that the generator is a \(d\)-monotone function. The class of \(d\)-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of \(d\)-dimensional \(\ell _{1}\)-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 R. E. Williamson [Duke Math. J. 23, 189–207 (1956; Zbl 0070.28501)] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform [see D. V. Widder, The Laplace transform. NY: Princeton Press (1946; JFM 67.0384.01)].
These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean 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 singular 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.
These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean 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 singular 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.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 60E05 | Probability distributions: general theory |
| 62E10 | Characterization and structure theory of statistical distributions |
Keywords:
d-monotone function; dependence ordering; frailty model; \(\ell_1\)-norm symmetric distribution; Laplace transform; stochastic simulation; Williamson d-transformReferences:
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