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Dependence measures for extreme value analyses. (English) Zbl 0972.62030
Let $$(X, Y)$$ be a random vector and let $$(U, V)$$ be the transformation of $$(X, Y)$$ to uniform margins. Let $\chi(u) = 2 - {\log Pr(U<u, V<v) \over \log Pr(U<u)} \sim Pr(V> v\mid U>u), \quad u \to 1.$ The authors consider the value $$\chi =\lim\limits_{u\to 1} \chi(u)$$, where $$\chi$$ is the probability of one variable being extreme given that the other is extreme. In the case $$\chi =0$$ the variables are said to be asymptotically independent. The authors define a new quantity $$\bar \chi = \lim\limits_{u\to 1} \bar\chi(u)$$, where $\bar\chi(u)=2\log Pr(U>u)[\log Pr(U>u, V>v)]^{-1}-1.$ If $$\chi =0$$ for the general class of distributions, then all members of this class are asymptotically independent. But at finite levels they have quite different degrees of dependence. The quantity $$\bar\chi$$ gives a suitable measure of extremal dependence within this class. The connections between $$\chi$$ and $$\bar\chi$$ are established. The authors study properties of $$\chi$$ and $$\bar\chi$$, consider many examples and discuss connections between this paper and general models for multivariate extremes.

##### MSC:
 62G32 Statistics of extreme values; tail inference 60G70 Extreme value theory; extremal stochastic processes
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