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Asymptotic independence and support detection techniques for heavy-tailed multivariate data. (English) Zbl 1448.62074
Summary: One of the central objectives of modern risk management is to find a set of risks where the probability of multiple simultaneous catastrophic events is negligible. That is, risks are taken only when their joint behavior seems sufficiently independent. This paper aims to identify asymptotically independent risks by providing tools for describing dependence structures of multiple risks when the individual risks can obtain very large values.
The study is performed in the setting of multivariate regular variation. We show how asymptotic independence is connected to properties of the support of the angular measure and present an asymptotically consistent estimator of the support. The estimator generalizes to any dimension \(N \geq 2\) and requires no prior knowledge of the support. The validity of the support estimate can be rigorously tested under mild assumptions by an asymptotically normal test statistic.
MSC:
62H12 Estimation in multivariate analysis
62E20 Asymptotic distribution theory in statistics
62G32 Statistics of extreme values; tail inference
62G05 Nonparametric estimation
60G70 Extreme value theory; extremal stochastic processes
60G57 Random measures
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