## Copula and semicopula transforms.(English)Zbl 1078.62055

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.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 60E05 Probability distributions: general theory 62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text: