Best-possible bounds on the sets of copulas and quasi-copulas with given curvilinear sections

• Published in: Fuzzy sets and systems Vol. 441; pp. 335 - 365
• Main Authors: Zou, Wei, Sun, Liulei, Xie, Jiehua
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 05.08.2022
• Subjects: Best-possible bound; Copula; Curvilinear section; Quasi-copula; Best-possible bound; Quasi-copula; Curvilinear section; Copula
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2021.12.008

In this paper, we consider the (quasi-)copulas with a given curvilinear section, and then establish the best-possible bounds on the set of quasi-copulas with a common curvilinear section. The lower bound is shown to be the best-possible lower bound as well for the set of copulas with the given curvilinear section. A sufficient and necessary condition guaranteeing the upper bound to be a copula is provided, and it is shown that when the upper bound is a copula, it is the best-possible upper bound for the set of copulas with the given curvilinear section. The best-possible upper bound on the set of copulas which have a common curvilinear section and are symmetrical with respect to the given curve is also provided. Some properties of the best-possible bound copulas with the given curvilinear section are also discussed, such as the statistical characteristic, the concordance order, the nonparametric measure of association and the tail dependence.