We describe a class of bivariate copulas having a fixed diagonal section. The obtained class contains both the Fréchet upper and lower bounds and it allows to describe non-trivial tail dependence coefficients along both the diagonals of the unit square.
The class of componentwise concave copulas is considered, with particular emphasis on its closure under some constructions of copulas (e.g., ordinal sum) and its relations with other classes of copulas characterized by some notions of concavity and/or convexity. Then, a sharp upper bound is given for the L∞-measure of non-exchangeability for copulas belonging to this class.
n this paper we study the set of copulas for which both a horizontal section and a vertical section have been given. We give a general construction for copulas of this type and we provide the lower and upper copulas with these sections. Symmetric copulas with given horizontal section are also discussed, as well as copulas defined on a grid of the unit square. Several examples are presented.
We give a representation of the class of all n-dimensional copulas such that, for a fixed m∈N, 2≤m<n, all their m-dimensional margins are equal to the independence copula. Such an investigation originated from an open problem posed by Schweizer and Sklar.
We investigate two constructions that, starting with two bivariate copulas, give rise to a new bivariate and trivariate copula, respectively. In particular, these constructions are generalizations of the ∗-product and the ⋆-product for copulas introduced by Darsow, Nguyen and Olsen in 1992. Some properties of these constructions are studied, especially their relationships with ordinal sums and shuffles of Min.
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.