Dependence and order in families of Archimedean copulas

The copula for a bivariate distribution function H(x, y) with marginal distribution functions F(x) and G(y) is the function C defined by H(x, y)=C(F(x), G(y)). C is called Archimedean if C(u, v)=(phi(-1)(phi(u)+phi(v)), where phi is a convex decreasing continuous function on (0, 1) with (phi(1)=0. A copula has lower tail dependence if C(u, u)/u converges to a constant y in (0, 1] as u-->0(+); and has upper tail dependence if <(C)over cap(u, u)>/(1-u) converges to a constant delta in (0, 1) as u-->1(-) where (C) over cap denotes the survival function corresponding to C. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values of gamma and delta, 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. (C) 1997 Academic Press.