Diagonal copulas. (English) Zbl 0906.60021

Nelsen, Roger B.; Fredricks, Gregory A.

Beneš, Viktor (ed.) et al., Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 121-128 (1997).

Summary: A diagonal copula has the form \(K(u,v)= \min (u,v,(1/2) [\delta(u) +\delta (v)])\) where \(\delta\) is any function satisfying (i) \(\delta (1)=1\); (ii) \(0\leq \delta (t_2)- \delta (t_1)\leq 2 (t_2- t_1)\) for all \(t_1\), \(t_2\) in \([0,1]\) with \(t_1\leq t_2\); and (iii) \(\delta (t)\leq t\) for all \(t\in [0,1]\). A diagonal copula is an ordinary sum of Min and what we call quasi-hairpin copulas, and conversely. Diagonal copulas are symmetric, singular and extremal. We relate them to shuffles of Min and copulas with hairpin support, and prove the following characterization theorem: Suppose \(X\) and \(Y\) are continuous random variables with copula \(C\) and a common marginal distribution function. Then the joint distribution function of \(\max (X,Y)\) and \(\min (X,Y)\) is the Fréchet upper bound if and only if \(C\) is a diagonal copula.
For the entire collection see [Zbl 0885.00054].

MSC:

60E99

Distribution theory