Curvilinear patchwork constructions of (quasi-)copulas with given curvilinear sections

In this paper, we focus on constructions of (quasi-)copulas with a given curvilinear section by the curvilinear patchwork operation. It is shown that the curvilinear patchwork of any two quasi-copulas with a common curvilinear section is always a quasi-copula, and some sufficient conditions for the curvilinear patchwork of two copulas with a common curvilinear section to be a copula are also provided. The density, the singular component and tail dependence coefficients of copulas constructed by the curvilinear patchwork operation are discussed. The concept of a simple curvilinear section is introduced, and an elementary way of creating copulas with simple curvilinear sections is developed by the curvilinear patchwork construction. Best-possible bounds for the set of copulas with a common simple curvilinear section are given. By the curvilinear patchwork construction, two generalized bound copulas with given curvilinear sections and a new extension of the Farlie-Gumbel-Morgenstern family of copulas are also proposed.