×

A remark on associative copulas. (English) Zbl 1010.60014

Let \(I=[0,1]\). Copulas are distribution functions on \(I^2\) with uniform marginals. Assume that \(\oplus \) is a continuous associative operation in \([0,a]\), \(a\in [0,\infty ]\), such that \(t\oplus 0= 0\oplus t=t\), \(t\oplus a=a\oplus t=a\) for all \(t\in [0,a]\). A function \(\psi :[0,a]\to R\) is called \(\oplus \)-convex if \(\psi (r\oplus t)-\psi (r)\leq \psi (s\oplus t) -\psi (s)\) for every \(r\leq s\) and any \(t\). The main result of the paper can be formulated as follows. Let \(\varphi :I\to [0,a]\) be a strictly decreasing continuous surjection. Define \(C(x,y) =\varphi ^{-1}(\varphi (x)\oplus \varphi (y))\). Then \(C\) is a copula if and only if \(\varphi ^{-1}\) is \(\oplus \)-convex.

MSC:

60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
PDF BibTeX XML Cite
Full Text: EuDML