Dependence and Order in Families of Archimedean Copulas

The copula for a bivariate distribution functionH(x, y) with marginal distribution functionsF(x) andG(y) is the functionCdefined byH(x, y)=C(F(x), G(y)).Cis called Archimedean ifC(u, v)=ϕ−1(ϕ(u)+ϕ(v)), whereϕis a convex decreasing continuous function on (0, 1] withϕ(1)=0. A copula has lower tail dependence ifC(u, u)/uconverges to a constantγin (0, 1] asu→0+; and has upper tail dependence ifC(u, u)/(1−u) converges to a constantδin (0, 1] asu→1−whereCdenotes the survival function corresponding toC. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values ofγandδ, and present extensions to higher dimensions. We also investigate limiting cases and the concordance ordering of these families. In the process, we present answers to two open problems posed by Joe (1993,J. Multivariate Anal.46262–282).