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Dependence and order in families of Archimedean copulas. (English) Zbl 0883.62049
Summary: 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 \(C(u,v)= \varphi^{-1} (\varphi(u)+ \varphi(v))\), where \(\varphi\) is a convex decreasing continuous function on \((0,1]\) with \(\varphi(1)=0\). A copula has lower tail dependence if \(C(u,u)/u\) converges to a constant \(\gamma\) in \((0,1]\) as \(u\to 0^+\); and has upper tail dependence if \(\overline{C}(u,u)/ (1-u)\) converges to a constant \(\delta\) in \((0,1]\) as \(u\to 1^-\), where \(\overline{C}\) denotes the survival function corresponding to \(C\).
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 H. Joe [ibid. 46, No. 2, 262-282 (1993; Zbl 0778.62045)].

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
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