Diagonal copulas.(English)Zbl 0906.60021

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:

 6e+100 Distribution theory