On a family of copulas constructed from the diagonal section. (English) Zbl 1098.60016

Durante, F.; Mesiar, R.; Sempi, C.

Soft Comput. 10, No. 6, 490-494 (2006).

Summary: We characterize the class of copulas that can be constructed from the diagonal section by means of the functional equation \(C (x, y)+| x - y |= C (x \vee y, x \vee y)\), for all \((x, y)\) in the unit square such that \(C (x, y)>0\). Some statistical properties of this class are given.

MSC:

60E05	Probability distributions: general theory
26B35	Special properties of functions of several variables, Hölder conditions, etc.
62H20	Measures of association (correlation, canonical correlation, etc.)

Keywords:

Aggregation operators; Measures of association

References:

[1]	Alsina, C.; Nelsen, RB; Schweizer, B., On the characterization of a class of binary operations on distribution functions, Statist Probab Lett, 17, 85-89 (1993) · Zbl 0798.60023
[2]	Calvo T, Kolesárová A, Komorníková M, Mesiar R (2002) Aggregation operators: properties, classes and construction methods. In: Calvo T, Mesiar R, Mayor G (eds), Aggregation operators. New trends and applications. Physica, Heidelberg, pp 3-106 · Zbl 1039.03015
[3]	Bassan, B.; Spizzichino, F., Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes, J Multivar Anal, 93, 313-339 (2005) · Zbl 1070.60015
[4]	Bertino, S., Sulla dissomiglianza tra mutabili cicliche, Metron, 35, 53-88 (1977) · Zbl 0437.62026
[5]	Durante, F.; Sempi, C., Semicopulæ. Kybernetika, 41, 315-328 (2005) · Zbl 1249.26021
[6]	Dunford N, Schwartz JT (1958) Linear operators. Part I: General theory. Wiley, New York · Zbl 0084.10402
[7]	Fredricks GA, Nelsen RB (1997) Copulas constructed from diagonal sections. In: Beneš V, Štěpán J (eds), Distributions with given marginals and moment problems. Kluwer, Dordrecht, pp 129-136 · Zbl 0906.60022
[8]	Fredricks GA, Nelsen RB (2003) The Bertino family of copulas. In: Cuadras CM, Fortiana J, Rodrí guez Lallena JA (eds), Distributions with given marginals and Statistical Modelling. Kluwer, Dordrecht, pp 81-91 · Zbl 1135.62334
[9]	Genest C, Quesada Molina JJ, Rodríguez Lallena JA, Sempi C (1999) A characterization of quasi-copulas. J Multivar Anal 69:193-205 · Zbl 0935.62059
[10]	Klement EP, Kolesárová A (2004) 1-Lipschitz aggregation operators, quasi-copulas and copulas with given diagonals. In: López Diaz M, Gil MA, Grzegorzewski P, Hryniewicz O, Lawry J (eds), Soft methodology and random information systems, advances in soft computing, pp 205-211 · Zbl 1071.62048
[11]	Klement, EP; Kolesárová, A., Extension to copulas and quasi-copulas as special 1-Lipschitz aggregation operators, Kybernetika, 41, 3 (2005) · Zbl 1249.60017
[12]	Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer, Dordrecht · Zbl 0972.03002
[13]	Klement, EP; Mesiar, R.; Pap, E., Invariant copulas, Kybernetika, 38, 275-285 (2002) · Zbl 1264.62045
[14]	Klement, EP; Mesiar, R.; Pap, E., Measure-based aggregation operators, Fuzzy Sets Syst, 142, 3-14 (2004) · Zbl 1046.28011
[15]	Mayor, G.; Torrens, J., On a family of t-norms, Fuzzy Sets Syst, 41, 161-166 (1991) · Zbl 0739.39006
[16]	Nelsen RB, Fredricks GA (1997) Diagonal copulas. In: Beneš V, Štěpán J (eds), Distributions with given marginals and moment problems. Kluwer, Dordrecht, pp 121-128 · Zbl 0906.60021
[17]	Nelsen RB (1999) An introduction to copulas. Lecture Notes in Statistics. Springer, New York · Zbl 0909.62052
[18]	Nelsen RB, Quesada Molina JJ, Rodrí guez Lallena JA, Úbeda Flores M (2004) Best-possible bounds on sets of bivariate distribution functions. J Multivar Anal 90:348-358 · Zbl 1057.62038
[19]	Schweizer B, Sklar A (1983) Probabilistic metric spaces. Elsevier, New York · Zbl 0546.60010
[20]	Sklar, A., Fonctions de répartition à n dimensions et leurs marges, Publ Inst Statist Univ Paris, 8, 229-231 (1959) · Zbl 0100.14202