Probability Theory
The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas
Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 583-586.

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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.09.026
Nelsen, Roger B. 1; Úbeda Flores, Manuel 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
@article{CRMATH_2005__341_9_583_0,
     author = {Nelsen, Roger B. and \'Ubeda Flores, Manuel},
     title = {The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--586},
     publisher = {Elsevier},
     volume = {341},
     number = {9},
     year = {2005},
     doi = {10.1016/j.crma.2005.09.026},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.09.026/}
}
TY  - JOUR
AU  - Nelsen, Roger B.
AU  - Úbeda Flores, Manuel
TI  - The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 583
EP  - 586
VL  - 341
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.09.026/
DO  - 10.1016/j.crma.2005.09.026
LA  - en
ID  - CRMATH_2005__341_9_583_0
ER  - 
%0 Journal Article
%A Nelsen, Roger B.
%A Úbeda Flores, Manuel
%T The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas
%J Comptes Rendus. Mathématique
%D 2005
%P 583-586
%V 341
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.09.026/
%R 10.1016/j.crma.2005.09.026
%G en
%F CRMATH_2005__341_9_583_0
Nelsen, Roger B.; Úbeda Flores, Manuel. 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. http://www.numdam.org/articles/10.1016/j.crma.2005.09.026/

[1] Alsina, C.; Nelsen, R.B.; Schweizer, B. On the characterization of a class of binary operations on distribution functions, Statist. Probab. Lett., Volume 17 (1993), pp. 85-89

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

[3] Genest, C.; Quesada Molina, J.J.; Rodríguez Lallena, J.A.; Sempi, C. A characterization of quasi-copulas, J. Multivariate Anal., Volume 69 (1999), pp. 193-205

[4] Nelsen, R.B. An Introduction to Copulas, Springer-Verlag, Berlin/New York, 1999

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

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

Cited by Sources: