Dependence and Order in Families of Archimedean Copulas

✍ Scribed by Roger B. Nelsen

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 973 KB
• Volume: 60
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.1646

✦ Synopsis

The copula for a bivariate distribution function H(x, y) with marginal distribution functions F (x) and G( y) is the function

, where . is a convex decreasing continuous function on (0, 1] with .(1)=0. A copula has lower tail dependence if C(u, u)Âu converges to a constant # in (0, 1] as u Ä 0 + ; and has upper tail dependence if C (u, u)Â(1&u) converges to a constant $ in (0, 1] as u Ä 1 & where C denotes the survival function corresponding to C. In this paper we develop methods for generating families of Archimedean copulas with arbitrary values of # and $, 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. 46 262 282).

1997 Academic Press two random variables X and Y with marginal distribution functions F and G, respectively. A copula is itself a bivariate distribution function with margins uniform on [0, 1]. For a general discussion of copulas and their properties, see Chapter 6]. We will review the concepts of lower and upper tail dependence in copulas in Section 2.