DUCS copulas

Copulas [18] link univariate marginal distribution functions into a joint distribution function of the corresponding random vector. In this paper we will deal with bivariate copulas only. Recall that a function C : [0, 1] → [0, 1] is a (bivariate) copula whenever it is grounded, C(x, y) = 0 whenever 0 ∈ {x, y}, it has neutral element 1, C(x, y) = x∧y, whenever 1 ∈ {x, y} and it is 2-increasing, C(x+ ǫ, y+ δ)−C(x, y+ δ) ≥ C(x+ǫ, y)−C(x, y) for all x, y, ǫ, δ ∈ [0, 1] such that x+ǫ, y+δ ∈ [0, 1]. Three basic copulas Π,M,W given by Π(x, y) = xy, M(x, y) = x∧y, W (x, y) = (x+y−1)∨0, express the independence, total comonotone dependence (Y = φ(X) for an increasing function φ) and total countermonotone dependence (Y = η(X) for a decreasing function η) of the univariate random variables X and Y , respectively. For modelling purposes, the knowledge of a large class of copulas is required. Thus several parametric classes of copulas have been introduced. For an overview we recommend monographs [8, 16]. It seems so that the most prominent class of copulas is the class of Archimedean copulas together with their M -ordinal sums. For more details we recommend [16, 19]. Note only that by Cf we denote an Archimedean copula Cf : [0, 1] 2 → [0, 1] given by