Diagonal Copulas

Abstract

A diagonal copula has the form K(u,υ) = min(u, υ,(1/2)[δ(u) + δ(υ)]) where δ is any function satisfying (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 ∈ [0,1]. A diagonal copula is an ordinal 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.