The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas

Roger B. Nelsen; Manuel Úbeda Flores

doi:10.1016/j.crma.2005.09.026

Probability Theory

The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas

Presented by: Paul Deheuvels

Roger B. Nelsen ¹; Manuel Úbeda Flores ²

¹ Department of Mathematical Sciences, Lewis & Clark College, 0615 S.W. Palatine Hill Road, Portland, OR 97219, USA
² Departamento de Estadística y Matemática Aplicada, Universidad de Almería, Carretera de Sacramento s/n, La Cañada de San Urbano, 04120 Almería, Spain

Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 583-586.

Abstract
Résumé

In this Note we show that the set of quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind–MacNeille completion of the set of copulas. Consequently, any set of copulas sharing a particular statistical property is guaranteed to have pointwise best-possible bounds within the set of quasi-copulas.

Dans cette Note, nous montrons que l'ensemble des quasi-copules est un treillis complet, qui est isomorphe au sens de l'ordre à la complétion de Dedekind–MacNeille de l'ensemble des copules. En conséquence, tout ensemble de copules qui possède une propriété statistique particulière est assuré de réaliser les meilleures bornes ponctuelles parmi l'ensemble des quasi-copules.

Received: 2005-07-20
Accepted: 2005-09-15
Published online: 2005-10-11

DOI: 10.1016/j.crma.2005.09.026

Author's affiliations:

Roger B. Nelsen ¹; Manuel Úbeda Flores ²

¹ Department of Mathematical Sciences, Lewis & Clark College, 0615 S.W. Palatine Hill Road, Portland, OR 97219, USA
² Departamento de Estadística y Matemática Aplicada, Universidad de Almería, Carretera de Sacramento s/n, La Cañada de San Urbano, 04120 Almería, Spain

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Roger B. Nelsen; Manuel Úbeda Flores. The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas. Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 583-586. doi : 10.1016/j.crma.2005.09.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.026/

References
Cited by

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[2] B.A. Davey; H.A. Priestley Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002

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[5] R.B. Nelsen; J.J. Quesada Molina; J.A. Rodríguez Lallena; M. Úbeda Flores Best-possible bounds on sets of bivariate distribution functions, J. Multivariate Anal., Volume 90 (2004), pp. 348-358

[6] A. Sklar Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, Volume 8 (1959), pp. 229-231

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