The Annals of Statistics

Estimation of extreme risk regions under multivariate regular variation

Juan-Juan Cai, John H. J. Einmahl, and Laurens de Haan

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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.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1803-1826.

First available in Project Euclid: 25 July 2011

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Zentralblatt MATH identifier

Primary: 62G32: Statistics of extreme values; tail inference 62G05: Estimation 62G07: Density estimation
Secondary: 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems

Extremes level set multivariate quantile rare event spectral density tail dependence


Cai, Juan-Juan; Einmahl, John H. J.; de Haan, Laurens. Estimation of extreme risk regions under multivariate regular variation. Ann. Statist. 39 (2011), no. 3, 1803--1826. doi:10.1214/11-AOS891.

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