From Archimedean to Liouville copulas

• Published in: Journal of multivariate analysis Vol. 101; no. 8; pp. 1772 - 1790
• Main Authors: McNeil, Alexander J., Nešlehová, Johanna
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01-09-2010; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: [formula omitted]-norm symmetric distribution; Archimedean copula; Archimedean copula Simplex distribution l1-norm symmetric distribution Liouville distribution Kendall's tau Williamson d-transform Laplace transform Stochastic ordering Dependence ordering Stochastic simulation; Dependence ordering; Distribution theory; Exact sciences and technology; Kendall’s tau; Laplace transform; Laplace transforms; Liouville distribution; Mathematics; Multivariate analysis; Nonparametric inference; Probability and statistics; Probability distribution; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Simplex distribution; Statistics; Stochastic ordering; Stochastic simulation; Studies; Williamson [formula omitted]-transform; secondary; Archimedean copula; Williamson d -transform; Dependence ordering; Stochastic ordering; Stochastic simulation; ℓ 1 -norm symmetric distribution; Liouville distribution; Laplace transform; Simplex distribution; Kendall’s tau; primary; 62H05; Stochastic order; Kendall's tau; Multivariate analysis; secondary 60E05; Symmetric law; Survival function; Distribution function; Ordering; Kendall tau; 62H20; norm symmetric distribution; Random variable; Correlation coefficient; Censored data; Copulas; Symmetric function; Williamson d-transform; l; Survival; Statistical method; primary 62E10; Rank correlation; Laplace transformation; L1 norm
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2010.03.015

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.