Copula and semicopula transforms (Q2570970)

The aim of this paper is to obtain interesting characterizations of copulas, quasi-copulas and semi-copulas on the basis of a certain type of transformations. The authors consider, for every copula \(C\) and quasi-copula \(Q\), the transformations \[ C_h(x,y)=h^{[-1]}(C(h(x),h(y)))\quad\text{and }\quad Q_h(x,y)=h^{[-1]}(Q(h(x),h(y))), \] respectively, with \(x, y \in [0, 1]\). The function \(h\) is assumed to be continuous and strictly increasing on [0, 1], with \(h(1) = 1\). Let \(H\) denote the set of functions \(h\) with these properties. The function \(h^{[-1]}: [0, 1]\to [0, 1]\), called the pseudo-inverse of \(h\), is defined as follows: \[ h^{[-1]}(t)=\begin{cases} h^{-1}(t),&\text{for }h(0)\leq t\leq1;\\0,&\text{for }0\leq t\leq h(0).\end{cases} \] Section 2 of the paper is studying the properties of semi-copulas via the transform \(C_h(x, y)\). The same type of transformations is used to characterize the copulas in Section 3 and quasi-copulas in Section 4. The main results of the paper are expressed as theorems of the following type: Theorem 1. For each function \(h\in H\), the following statements are equivalent: \(h\) is concave; for every copula \(C\), the transform \(C_h\) (defined above) is a copula, too. Theorem 2. For each function \(h\in H\), the following statements are equivalent: \(h\) is concave; for every quasi-copula \(Q\), the transform \(Q_h\) (defined above) is a quasi-copula, too. These characterizations of copulas, quasi-copulas, and semi-copulas may represent formal tools for further functional and algebraic properties of these concepts, in relationship to other approaches based on \(t\)-norms, \(t\)-semi-norms, binary aggregation operators, etc.