Estimation of extreme risk regions under multivariate regular variation
Juan-Juan Cai, John H. J. Einmahl, Laurens de Haan
The Annals of Statistics · 2011 · Volume 39 · Issue 3
Abstract
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.
Keywords
60F05, 60G70, 62G05, 62G07, 62G32, Extremes, Level set, Multivariate quantile, Rare event, Spectral density, tail dependence
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