From Archimedean to Liouville copulas

Alexander J. McNeil, Johanna Nešlehová*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (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.

Original languageEnglish
Pages (from-to)1772-1790
Number of pages19
JournalJournal of Multivariate Analysis
Issue number8
Publication statusPublished - Sept 2010


  • Archimedean copula
  • Dependence ordering
  • Kendall's tau
  • Laplace transform
  • Liouville distribution
  • Primary
  • Secondary
  • Simplex distribution l-norm symmetric distribution
  • Stochastic ordering
  • Stochastic simulation
  • Williamson d-transform

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