From Archimedean to Liouville copulas

Authors:
Alexander J. McNeil;Johanna Nešlehová
Affiliations:
Maxwell Institute for Mathematical Sciences, Edinburgh, UK;Department of Mathematics and Statistics, McGill University, Montréal, Canada
Venue:
Journal of Multivariate Analysis
Year:
2010

Citing 6
Cited 4

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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 Neslehova (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.