Dependence measures for extreme value analyses. (English) Zbl 0972.62030

Coles, Stuart; Heffernan, Janet; Tawn, Jonathan

Extremes 2, No. 4, 339-365 (1999).

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.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

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

Keywords:

point processes; asymptotic independence; bivariate extreme value distributions; copulas; dependence measures