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A directory of coefficients of tail dependence. (English) Zbl 0979.62040
Let \((X,Y)\) be an \(R^2\) random vector with unit Fréchet margins and \[ \Pr\{X>t,Y>t\}\sim L(t)/\Pr\{X>t\}^{1/\eta} \] for large \(t\), \(L(t)\) slowly varying as \(t\to\infty\) and \(\eta\in[0,1]\). Then \(\eta\) is called the coefficient of tail dependence. (The independence of \(X\) and \(Y\) corresponds to \(\eta=1/2\), \(L(t)=1\)). The coefficient \(\eta\) is connected with the \(\bar\chi\) dependence measure introduced by S. Coles, J. Heffernan and J. Tawn [Extremes 2, No. 4, 339-365 (1999; Zbl 0972.62030)] and based on the copula representation of the d.f. \(F_{XY}\): \(F_{XY}(x,y)=C(F_X(x),F_Y(y))\) (here \(C\) is some function, uniquely defined by this relation). Then \[ \bar\chi=\lim_{u\to 1} 2\log(1-u)\log^{-1}(1-u-v-C(u,v)) \] and \(\bar\chi=2\eta-1\). The paper presents \(\eta\), \(L(t)\) and copula representations for different parametric families of distributions such as Pareto, extreme value, Gaussian, Frank, Morgenstern, Raftery, etc.

62G32 Statistics of extreme values; tail inference
60E99 Distribution theory
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