Set Inference for Multivariate Extreme Quantile Region under Regular Variation

Abstract

Consider the extreme quantile region in arbitrary dimension as the half-space depth trimmed region at a very small probability level, which coincides with the envelope of the directional extreme quantiles. The classical asymptotic confidence regions significantly undercover the population set in finite samples, even when the sample size is in thousands. We propose a second-order correction under regular variation base on the joint convergence of the properly normalized second-order residual process of the extreme directional quantile estimator and the Hill estimator of the tail index. Our approach relies on a dual relationship between the set-valued estimator and its support function, and recognizes the shape estimation error of our quantile region. Our second-order confidence regions improve the coverage towards correct levels in simulation studies. We offer finance applications in a six-dimensional international stock market dataset. Extensions beyond the half-space depth are discussed.

Keywords: extreme value statistics, level set, multivariate quantile, confidence region, outlier detection, rare event, tail dependence