Estimation of extreme depth-based quantile regions

Consider the extreme quantile region induced by the half-space depth function HD of the form Q={x∈R^d ∶HD(x,P)≤β}, such that PQ = p for a given, very small p>0. Since this involves extrapolation outside the data cloud, this region can hardly be estimated through a fully non-parametric procedure. Using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result. A simulation study clearly demonstrates the good performance of our estimator. We use the procedure for risk management by applying it to stock market returns.