Cai, J.J., Einmahl, JohnH.J. and De Haan, L. (2011). Estimation of extreme risk regions under multivariate regular variation Annals of Statistics, 39(3):1803--1826.
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Affiliated author
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Publication year2011
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JournalAnnals of Statistics
When considering d possibly dependent random variables, one is often interested in extreme risk regions, with very small probability p. We consider risk regions of the form \{z ε ℝd : f (z) ≤ β\}, where f is the joint density and β a small number. Estimation of such an extreme risk region is difficult since it contains hardly any or no data. Using extreme value theory, we construct a natural estimator of an extreme risk region and prove a refined form of consistency, given a random sample of multivariate regularly varying random vectors. In a detailed simulation and comparison study, the good performance of the procedure is demonstrated. We also apply our estimator to financial data.