Copulas Constructed from Diagonal Sections

Abstract

If C is a copula, then its diagonal section is the function δ given by δ(t) = C(t, t). It follows that (i) δ(1) = 1; (ii) 0 ≤ δ(t 2) - δ(t 1) ≤ 2(t 2 - t 1) for all t 1, t 2 in [0,1] with t 1 ≤ t 2; and (iii) δ(t) ≤ t for all t in [0,1]. If δ is any function satisfying (i)-(iii), does there exist a copula C whose diagonal section is δ? We answer this question affirmatively by constructing copulas we call diagonal copulas: K(u, υ) = min(u, υ, (1/2)[δ(u) + δ(υ)]). 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 posed by H. Joe.