Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Archimedean copula; associativity in the sense of Post; $n$-dimensional copula
Archimedean copula; associativity in the sense of Post; $n$-dimensional copula
Summary:
The associativity of $n$-dimensional copulas in the sense of Post is studied. These copulas are shown to be just $n$-ary extensions of associative 2-dimensional copulas with special constraints, thus they solve an open problem of R. Mesiar posed during the International Conference FSTA 2010 in Liptovský Ján, Slovakia.
References:
[1] Couceiro, M.: On two generalizations of associativity. In: Abstracts of FSTA 2010 (E. P. Klement et. al., eds.), Liptovský Ján 2010, p. 47.
[2] Fodor, J., Yager, R., Rybalov, A.: Structure of uninorms. Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 5 (1997), 411–427. DOI 10.1142/S0218488597000312 | MR 1471619
[3] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge 2009. MR 2538324 | Zbl 1196.00002
[4] Joe, H.: Multivariable Models and Dependence Concepts. Chapman & Hall, London 1997. MR 1462613
[5] Klement, E.-P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dortrecht 2000. MR 1790096 | Zbl 1010.03046
[6] Ling, C. H.: Representation of associative functions. Publicationes Mathematicae Debrecen 12 (1965), 189–212. MR 0190575
[7] McNeil, A.-J., Nešlehová, J.: Multivariate Archimedean copulas, $d$-monotone functions and $L_1$-norm symmetric distributions. Annals Statist. 37 (2009), 3059–3097. DOI 10.1214/07-AOS556 | MR 2541455
[8] Mesiar, R., Sarkoci, P.: Open problems posed at the 10th International Conference on Fuzzy Set Theory and Appl. (FSTA 2010), Liptovský Ján. Kybernetika 46 (2010), 585–598. MR 2722089
[9] Mesiar, R., Sempi, C.: Ordinal sums and idempotents of copulas. Aequationes Math. 79 (2010), 1–2, 39–52. DOI 10.1007/s00010-010-0013-6 | MR 2640277 | Zbl 1205.62063
[10] Moynihan, R.: Infinite $\tau _T$ products of distribution functions. J. Austral. Math. Soc. Ser. A 26 (1978), 227–240. DOI 10.1017/S1446788700011721 | MR 0511607
[11] Nelsen, R.-B.: An Introduction to Copulas. Second edition. Springer Science and Business Media, New York 2006. MR 2197664 | Zbl 1152.62030
[12] Post, E.-L.: Polyadic groups. Trans. Amer. Math. Soc. 48 (1940), 208–350. DOI 10.1090/S0002-9947-1940-0002894-7 | MR 0002894 | Zbl 0025.01201
[13] Sklar, A.: Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229–231. MR 0125600
Search
Partner of
