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Estimation of extreme risk regions under multivariate regular variation
Juan-Juan Cai, John H. J. Einmahl, Laurens de Haan · 2012-11-22 · via math.ST updates on arXiv.org

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 ${\mathbf{z}\in\mathbb{R}^d:f(\mathbf{z})\leqβ}$, 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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