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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.

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