Open Access
June 2011 Estimation of extreme risk regions under multivariate regular variation
Juan-Juan Cai, John H. J. Einmahl, Laurens de Haan
Ann. Statist. 39(3): 1803-1826 (June 2011). DOI: 10.1214/11-AOS891


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 ∈ 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.


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Juan-Juan Cai. John H. J. Einmahl. Laurens de Haan. "Estimation of extreme risk regions under multivariate regular variation." Ann. Statist. 39 (3) 1803 - 1826, June 2011.


Published: June 2011
First available in Project Euclid: 25 July 2011

zbMATH: 1221.62075
MathSciNet: MR2850221
Digital Object Identifier: 10.1214/11-AOS891

Primary: 62G05 , 62G07 , 62G32
Secondary: 60F05 , 60G70

Keywords: Extremes , Level set , Multivariate quantile , Rare event , Spectral density , tail dependence

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 3 • June 2011
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