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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.

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