Estimation of Extreme Depth-Based Quantile Regions

Consider the extreme quantile region, induced by the halfspace depth function HD, of the form Q = fx 2 Rd : HD(x; P) g, such that PQ = p for a given, very small p > 0. This region can hardly be estimated through a fully nonparametric procedure since the sample halfspace depth is 0 outside the convex hull of the data. 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.