# zbMATH — the first resource for mathematics

Orbital semilinear copulas. (English) Zbl 1186.62067
Summary: We introduce four families of semilinear copulas (i.e., copulas that are linear in at least one coordinate of any point of the unit square) of which the diagonal and opposite diagonal sections are given functions. For each of these families, we provide necessary and sufficient conditions under which given diagonal and opposite diagonal functions can be the diagonal and opposite diagonal sections of a semilinear copula belonging to that family. We focus particular attention on the family of orbital semilinear copulas, which are obtained by linear interpolation on segments connecting the diagonal and opposite diagonal of the unit square.

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H20 Measures of association (correlation, canonical correlation, etc.) 62H10 Multivariate distribution of statistics
Full Text:
##### References:
 [1] B. De Baets, H. De Meyer, and R. Mesiar: Asymmetric semilinear copulas. Kybernetika 43 (2007), 221-233. · Zbl 1136.62350 · www.kybernetika.cz · eudml:33853 [2] B. De Baets, H. De Meyer, and M. Úbeda-Flores: Opposite diagonal sections of quasi-copulas and copulas. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 17 (2009), 481-490. · Zbl 1183.62087 · doi:10.1142/S0218488509006108 [3] B. De Baets, H. De Meyer, and M. Úbeda-Flores: Constructing copulas with given diagonal and opposite diagonal sections. Comm. Statist. - Theory Methods, to appear. · Zbl 1215.62051 · doi:10.1080/03610920903480866 [4] F. Durante and P. Jaworski: Absolutely continuous copulas with given diagonal sections. Comm. Statist. - Theory Methods 37 (2008), 2924-2942. · Zbl 1292.60019 · doi:10.1080/03610920802050927 [5] F. Durante, A. Kolesárová, R. Mesiar, and C. Sempi: Copulas with given diagonal sections, novel constructions and applications. Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 15 (2007), 397-410. · Zbl 1158.62324 · doi:10.1142/S0218488507004753 [6] F. Durante, A. Kolesárová, R. Mesiar, and C. Sempi: Semilinear copulas. Fuzzy Sets and Systems 159 (2008), 63-76. · Zbl 1274.62108 · doi:10.1016/j.fss.2007.09.001 [7] F. Durante, R. Mesiar, and C. Sempi: On a family of copulas constructed from the diagonal section. Soft Computing 10 (2006), 490-494. · Zbl 1098.60016 · doi:10.1007/s00500-005-0523-7 [8] A. Erdely and J. M. González-Barrios: On the construction of families of absolutely continuous copulas with given restrictions. Comm. Statist. - Theory Methods 35 (2006), 649-659. · Zbl 1098.60017 · doi:10.1080/03610920500498758 [9] P. Jaworski and T. Rychlik: On distributions of order statistics for absolutely continuous copulas with applications to reliability. Kybernetika 44 (2008), 757-776. · Zbl 1180.60013 · www.kybernetika.cz · eudml:33963 [10] H. Joe: Multivariate Models and Dependence Concepts. Chapman & Hall, London 1997. · Zbl 0990.62517 [11] E. Klement and A. Kolesárová: Extension to copulas and quasi-copulas as special 1-Lipschitz aggregation operators. Kybernetika 43 (2005), 329-348. · Zbl 1249.60017 · www.kybernetika.cz · eudml:33757 [12] R. Nelsen: An Introduction to Copulas. Second edition. Springer, New York, 2006. · Zbl 1152.62030 [13] R. Nelsen and G. Fredricks: Diagonal copulas. Distributions with given Marginals and Moment Problems (V. Beneš and J. Štěpán, Kluwer Academic Publishers, Dordrecht 1977, pp. 121-127. [14] R. Nelsen, J. Quesada-Molina, J. Rodríguez-Lallena, and M. Úbeda-Flores: On the construction of copulas and quasi-copula with given diagonal sections. Insurance: Math. Econ. 42 (2008), 473-483. · Zbl 1152.60311 [15] A. Sklar: Fonctions de répartition à $$n$$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231. · Zbl 0100.14202 [16] M.-H. Zhang: Modelling total tail dependence along diagonals. Insurance: Math. Econ. 42 (2008), 73-80. · Zbl 1142.62097 · doi:10.1016/j.insmatheco.2007.01.002
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.