zbMATH — the first resource for mathematics

Constructing Archimedean copulas from diagonal sections. (English) Zbl 1242.62041
Summary: We introduce a family \(\mathcal F\) of functions called diagonal generators. These are convex functions with the properties of diagonal sections of archimedean copulas. We show that to each diagonal generator \(f\) there corresponds an Archimedean copula \(C^{f}\) with the asymptotic representation \[ C^{f}(u_{1},u_{2}) = \lim_{k\to \infty } f^{k} [f^{ - k}(u_{1})+f^{ - k}(u_{2}) - 1]. \] Moreover, the diagonal section of \(C^{f}\) equals \(f\).
We characterize archimedean copulas in terms of their asymptotic form. We construct a family \(\mathcal F_{F}\) of diagonal generators, induced by a regular distribution function \(F\). We study a differential equation (depending on a function parameter), whose solution is \(F\). We give four applications of diagonal generators: to concordance, quadrant dependence, measures of dependence and convergence of copulas.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text: DOI
[1] Charpentier, A.; Segers, J., Convergence of Archimedean copulas, Statist. probab. lett., 78, 412-419, (2008) · Zbl 1139.62303
[2] Durante, F.; Kolesárová, A.; Mesiar, R.; Sempi, C., Copulas with given diagonal sections: novel constructions and applications, Internat. J. uncertain. fuzziness knowledge-based systems, 10, 490-494, (2007)
[3] Durante, F.; Mesiar, R.; Sempi, C., On a family of copulas constructed from the diagonal section, Soft comput., 10, 490-494, (2006) · Zbl 1098.60016
[4] Fredricks, G.A.; Nelsen, R.B., Copulas constructed from diagonal sections, (), 129-136 · Zbl 0906.60022
[5] Genest, C.; MacKay, J., Copules archimédiennes et familles des lois bidimensionnelles dont LES marges sont données, Canad. J. statist., 14, 145-159, (1986) · Zbl 0605.62049
[6] Hutchinson, T.P.; Lai, C.D., Continuous bivariate distributions. emphasising applications, (1990), Rumsby Sci. Publ. Adelaide · Zbl 1170.62330
[7] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[8] Kuczma, M., ()
[9] Ling, C.H., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401
[10] McNeil, A.J.; Nešlehová, J., Multivariate Archimedean copulas, \(d\)-monotone functions and \(\ell_1\)-norm symmetric distributions, Ann. statist., 37, 3059-3097, (2009) · Zbl 1173.62044
[11] Nelsen, R.B., An introduction to copulas, (1999), Springer New York · Zbl 0909.62052
[12] Nelsen, R.B.; Quesada-Molina, J.J.; Rodríguez-Lallena, J.A.; Úbeda-Flores, M., On the constructions of copulas and quasi-copulas with given diagonal sections, Insurance math. econom., 42, 473-483, (2008) · Zbl 1152.60311
[13] Sungur, E.A.; Yang, Y., Diagonal copulas of Archimedean class, Comm. statist., 25, 1659-1676, (1996) · Zbl 0900.62339
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.