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archimax copula; copula; dependence function; generator of a dependence function
We construct two pairs $(\mathscr{A}^{[1]}_{F}, \mathscr{A}^{[2]}_{F})$ and $(\mathscr{A}^{[1]}_{\psi}, \mathscr{A}^{[2]}_{\psi})$ 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 $\mathbb\psi$. 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 ${\mathbb\alpha}_{\rm s}$ are symmetric dependence functions.
[1] Capéraà, P., Fougères, A. L., Genest, C.: Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 (2000), 30-49. DOI 10.1006/jmva.1999.1845 | MR 1747422 | Zbl 0978.62043
[2] Genest, C., MacKay, J.: Copules archimédiennes et familles des lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145-159. DOI 10.2307/3314660 | MR 0849869
[3] Gudendorf, G., Segers, J.: Extreme-value copulas. In: Copula Theory and Its Applications, Warsaw 2009, Lecture Notes in Statist. Proc. 198, Springer 2010, pp. 127-146.
[4] Gumbel, E. J.: Bivariate exponential distributions. J. Amer. Statist. Assoc. 55 (1960), 698-707. DOI 10.1080/01621459.1960.10483368 | MR 0116403 | Zbl 0099.14501
[5] Hürlimann, W.: Properties and measures of dependence for the archimax copula. Adv. Appl. Statist. 5 (2005), 125-143. MR 2204874
[6] Hutchinson, T. P., Lai, C. D.: Continuous Bivariate Distributions. Emphasising Applications. Rumsby Sci. Publ., Adelaide 1990. MR 1070715 | Zbl 1170.62330
[7] Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997. MR 1462613 | Zbl 0990.62517
[8] Nelsen, R. B.: An Introduction to Copulas. Springer, New York 1999. MR 1653203 | Zbl 1152.62030
[9] Pickands, J.: Multivariate extreme value distributions. Bull. Int. Statist. Inst. 49 (1981), 859-879. MR 0820979 | Zbl 0518.62045
[10] Wysocki, W.: When a copula is archimax. Statist. Probab. Lett. (2012), to appear.
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