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 con verges to a constant y in (0, 1] as u-->0(+); and has upper tail depen dence 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 corr esponding to C. In this paper we develop methods for generating famili es of Archimedean copulas with arbitrary values of gamma and delta, an d present extensions to higher dimensions. We also investigate limitin g cases and the concordance ordering of these Families. In the process , we present answers to two open problems posed by Joe. (C) 1997 Acade mic Press.