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Semicopulas: characterizations and applicability. (English) Zbl 1249.60016
Summary: We characterize some bivariate semicopulas and, among them, the semicopulas satisfying a Lipschitz condition. In particular, the characterization of harmonic semicopulas allows us to introduce a new concept of depedence between two random variables. The notion of multivariate semicopula is given and two applications in the theory of fuzzy measures and stochastic processes are given.

MSC:
60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
26B35 Special properties of functions of several variables, Hölder conditions, etc.
03E72 Theory of fuzzy sets, etc.
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[1] Alsina C., Frank M. J., Schweizer B.: Problems on associative functions. Aequationes Math. 66 (2003), 128-140 · Zbl 1077.39021 · doi:10.1007/s00010-003-2673-y
[2] Alsina C., Nelsen R. B., Schweizer B.: On the characterization of a class of binary operations on distribution functions. Statist. Probab. Lett. 17 (1993), 85-89 · Zbl 0798.60023 · doi:10.1016/0167-7152(93)90001-Y
[3] Axler S., Bourdon, P., Ramey W.: Harmonic Function Theory. (Graduate Texts in Mathematics 137.) Springer-Verlag, New York 2001 · Zbl 0959.31001 · doi:10.1007/b97238
[4] Baets B. De: Analytical solution methods for fuzzy relational equations. Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series (D. Dubois and H. Prade, Chapter 6, Vol. 1, Kluwer Academic Publishers, Dordrecht 2000, pp. 291-340 · Zbl 0970.03044
[5] Baets B. De, Meyer H. De: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152 (2005), 249-270 · Zbl 1114.91031 · doi:10.1016/j.fss.2004.11.002
[6] Baets B. De, Meyer H. De, Schuymer, B. De, Jenei S.: Cycle evaluation of transitivity of reciprocal relations. Soc. Choice Welfare. To appear · Zbl 1158.91338 · doi:10.1007/s00355-006-0093-3
[7] Schuymer B. De, Meyer, H. De, Baets B. De: On some forms of cycle-transitivity and their relation to commutative copulas. Proc. EUSFLAT-LFA Conference, Barcelona 2005, pp. 178-182
[8] Bassan B., Spizzichino F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93 (2005), 313-339 · Zbl 1070.60015 · doi:10.1016/j.jmva.2004.04.002
[9] Calvo T., Kolesárová A., Komorníková, M., Mesiar R.: Aggregation operators: properties, classes and construction methods. Aggregation Operators. New Trends and Applications (T. Calvo, R. Mesiar, and G. Mayor, Physica-Verlag, Heidelberg 2002, pp. 3-106 · Zbl 1039.03015
[10] Denneberg D.: Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht 1994 · Zbl 0968.28009
[11] Durante F.: What is a semicopula? In: Proc. AGOP - Summer School on Aggregation Operators, Lugano 2005, pp. 51-56
[12] Durante F.: A new class of symmetric bivariate copulas. Preprint n. 19, Dipartimento di Matematica E. De Giorgi, Lecce, 2005 · Zbl 1122.62039
[13] Durante F., Mesiar, R., Sempi C.: 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
[14] Durante F., Quesada-Molina J. J., Sempi C.: A generalization of the Archimedean class of bivariate copulas. Ann. Inst. Statist. Math. (2006), to appear · Zbl 1332.62171 · doi:10.1007/s10463-006-0061-9
[15] Durante F., Sempi C.: Semicopulæ. Kybernetika 41 (2005), 315-328 · Zbl 1249.26021 · www.kybernetika.cz · eudml:33756
[16] Genest C., Molina J. J. Quesada, Lallena J. A. Rodríguez, Sempi C.: A characterization of quasi-copulas. J. Multivariate Anal. 69 (1999), 193-205 · Zbl 0935.62059 · doi:10.1006/jmva.1998.1809
[17] Ricci R. Ghiselli, Mesiar R.: \(k\)-Lipschitz strict triangular norms. Proc. EUSFLAT-LFA Conference, Barcelona 2005, pp. 1307-1312 · Zbl 1116.39015
[18] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 · Zbl 1087.20041 · doi:10.1017/S1446788700008065
[19] Kolmogorov A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer-Verlag, Berlin 1933. Reprinted in: Foundations of the Theory of Probability. Chelsea, Bronxm NY 1950 · Zbl 0007.21601
[20] Mesiarová A.: \(k\)-Lipschitz aggregation operators. Proc. AGOP - Summer School on Aggregation Operators, Lugano 2005, pp. 89-92
[21] Mesiarová A.: Triangular norms and \(k\)-Lipschitz property. Proc. EUSFLAT-LFA Conference, Barcelona 2005, pp. 922-926
[22] Nelsen R. B.: An Introduction to Copulas. (Lecture Notes in Statistics 139.) Springer-Verlag, New York 1999 · Zbl 1152.62030 · doi:10.1007/978-1-4757-3076-0
[23] 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. 90 (2004), 348-358 · Zbl 1057.62038 · doi:10.1016/j.jmva.2003.09.002
[24] Nelsen R. B.: Copulas and quasi-copulas: an introduction to their properties and applications. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (E. P. Klement and R. Mesiar, Elsevier, Amsterdam 2005, pp. 391-413 · Zbl 1079.60021
[25] Rodríguez-Lallena J. A., Úbeda-Flores M.: A new class of bivariate copulas. Statist. Probab. Lett. 66 (2004), 315-325 · Zbl 1102.62054 · doi:10.1016/j.spl.2003.09.010
[26] Scarsini M.: Copulæ of capacities on product spaces. Distribution Functions with Fixed Marginals and Related Topics (L. Rüschendorf, B. Schweizer, and M. D. Taylor, Institute of Mathematical Statistics (Lecture Notes - Monograph Series Volume 28), Hayward 1996, pp. 307-318
[27] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North Holland, New York 1983. 2nd edition: Dover Publications, Mineola, New York 2005 · Zbl 0546.60010
[28] Stromberg K. R.: An Introduction to Classical Real Analysis. Chapman & Hall, London 1981 · Zbl 0454.26001
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