On the construction of families of absolutely continuous copulas with given restrictions. (English) Zbl 1098.60017

Erdely, Arturo; González-Barrios, José M.

Commun. Stat., Theory Methods 35, No. 4, 649-659 (2006).

Summary: The problem of constructing copulas with a given diagonal section has been studied by E. A. Sungur and Y. Yang [Commun. Stat., Theory Methods 25, No. 7, 1659–1676 (1996; Zbl 0900.62339)] and G. A. Fredricks and R. B. Nelsen [in: Distributions with given marginals and moment problems, 121–128 (1997; Zbl 0906.60021); ibid., 129–136 (1997; Zbl 0906.60022); and in: Distributions with given martingales and statistical modelling, 81–91 (2002)]. The results of Sungur and Yang are especially relevant because, among other results, they have proven that an Archimedean copula is characterized by its diagonal section. The results obtained by Fredricks and Nelsen allow one to build a singular copula with a given diagonal section. In all cases, the resulting copulas are symmetric. In this article, we provide a family of absolutely continuous copulas with a fixed diagonal, which can differ from another absolutely continuous copula almost everywhere with respect to Lebesgue measure. It is important to mention that the asymmetry in the proposed methodology is not an issue.

MSC:

60E05	Probability distributions: general theory
60A99	Foundations of probability theory

Keywords:

diagonal sections

Citations:

Zbl 0900.62339; Zbl 0906.60021; Zbl 0906.60022

References:

[1]	Fredricks G. A., Distributions with Given Marginals and Moment Problems pp 121– (1997)
[2]	Fredricks G. A., Distributions with Given Marginals and Moment Problems pp 129– (1997)
[3]	Fredricks G. A., Distributions with Given Marginals and Statistical Modelling pp 81– (2002)
[4]	Ling C.-H., Publ. Math. Debrechen 12 pp 189– (1965)
[5]	DOI: 10.1007/3-540-48236-9
[6]	Schweizer B., Probabilistic Metric Spaces (1983) · Zbl 0546.60010
[7]	DOI: 10.1080/03610929608831791