Kybernetika 48 no. 5, 977-987, 2012

Constructing families of symmetric dependence functions

Włodzimierz Wysocki


We construct two pairs $(\A^{[1]}_{F}, \A^{[2]}_{F})$ and $(\A^{[1]}_{\psi}, \A^{[2]}_{\psi})$ of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions $F$, and those of the second pair by elements $\psi$ of a specific function family $\bpsi$. We also show that all solutions of the differential equation $\frac{{\mathrm d}y}{{\mathrm d}u}=\frac{\alpha(u)}{u(1-u)}y$ for $\alpha$ in a certain function family $\balpha_{\rm s}$ are symmetric dependence functions.


copula, dependence function, archimax copula, generator of a dependence function




  1. P. Capéraà, A. L. Fougères and C. Genest: Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 (2000), 30-49.   CrossRef
  2. C. Genest and J. MacKay: Copules archimédiennes et familles des lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145-159.   CrossRef
  3. G. Gudendorf and J. Segers: Extreme-value copulas. In: Copula Theory and Its Applications, Warsaw 2009, Lecture Notes in Statist. Proc. 198, Springer 2010, pp. 127-146.   CrossRef
  4. E. J. Gumbel: Bivariate exponential distributions. J. Amer. Statist. Assoc. 55 (1960), 698-707.   CrossRef
  5. W. Hürlimann: Properties and measures of dependence for the archimax copula. Adv. Appl. Statist. 5 (2005), 125-143.   CrossRef
  6. T. P. Hutchinson and C. D. Lai: Continuous Bivariate Distributions. Emphasising Applications. Rumsby Sci. Publ., Adelaide 1990.   CrossRef
  7. H. Joe: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997.   CrossRef
  8. R. B. Nelsen: An Introduction to Copulas. Springer, New York 1999.   CrossRef
  9. J. Pickands: Multivariate extreme value distributions. Bull. Int. Statist. Inst. 49 (1981), 859-879.   CrossRef
  10. W. Wysocki: When a copula is archimax. Statist. Probab. Lett. (2012), to appear.   CrossRef