  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 ${\mathbf{z}\in\mathbb{R}^d:f(\mathbf{z})\leq\beta}$,
where f is the joint density and $\beta$ 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.
