Estimating extreme bivariate quantile regions

When simultaneously monitoring two possibly dependent, positive risks one is often interested in quantile regions with very small probability p. These extreme quantile regions contain hardly any or no data and therefore statistical inference is difficult. In particular when we want to protect ourselves against a calamity that has not yet occurred, we need to deal with probabilities p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) ∈ (0, ∞ )2: f(x, y) ≤ β}, where f, the joint density, is decreasing in both coordinates. Such a region has the property that it consists of the less likely points and hence that its complement is as small as possible. Using extreme value theory, we construct a natural, semiparametric estimator of such a quantile region and prove a refined form of consistency. A detailed simulation study shows the very good statistical performance of the estimated quantile regions. We also apply the method to find extreme risk regions for bivariate insurance claims.