Estimating Extreme Bivariate Quantile Regions

Abstract

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 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 take p < 1/n, with n the sample size. We consider quantile regions of the form {(x, y) Є (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. As an illustration, we compute the estimated quantile regions for simulated data sets.

Keywords: Density contour, extreme value, level set, multivariate quantile, rare event, semiparametric estimation, tail dependence

JEL Classification: C13, C14