Copula and semicopula transforms

The notion of copula was introduced by Sklar [24] who proved the theorem that now bears his name; it is commonly used in probability and statistics (see, for instance, [19, 22, 23]). Later, in order to characterize a class of operations on distribution functions that derive from operations on random variables defined on the same probability space, Alsina et al. [1] introduced the notion of quasi-copula (see also [12, 20, 27]). On the contrary, the notion of semicopula is recent [3, 8] and arises from a statistical application: the study of multivariate aging through the analysis of the Schur concavity of the survival function (see [2, 25]). Semicopulas generalize triangular norms (briefly t-norms), introduced by K. Menger in order to extend the triangle inequality from the setting of metric spaces to probabilistic metric spaces, and successfully used in probability theory, mathematical statistics, and fuzzy logic [15, 22]. We refer to our paper [8] for the properties of semicopulas. Here we recall that a semicopula is a function S : [0,1]2 → [0,1] that satisfies the following two conditions: ∀x in [0,1] S(x,1) = S(1,x) = x,