Copulas constructed from diagonal sections. (English) Zbl 0906.60022

Fredricks, Gregory A.; Nelsen, Roger B.

Beneš, Viktor (ed.) et al., Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 129-136 (1997).

Summary: If \(C\) is a copula, then its diagonal section is the function \(\delta\) given by \(\delta(t) =C(t,t)\). It follows that (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) \(\delta (t)\leq 1\) for all \(t\) in \([0,1]\). If \(\delta\) is any function satisfying (i)–(iii), does there exist a copula \(C\) whose diagonal section is \(\delta\)? We answer this question affirmatively by constructing copulas we call diagonal copulas: \(K(u,v)= \min (u,v,(1/2) [\delta (u)+ \delta(v)])\). We also present examples, investigate some dependence properties of random variables which have diagonal copulas, and answer an open question about tail dependence in bivariate copulas by H. Joe [J. Multivariate Anal. 46, No. 2, 262-282 (1993; Zbl 0778.62045)].
For the entire collection see [Zbl 0885.00054].

MSC:

60E99

Distribution theory

Citations:

Zbl 0778.62045