On the estimation of extreme directional multivariate quantiles

TL;DR

This paper introduces a new framework for estimating directional multivariate quantiles at extreme levels in multivariate extreme value theory, including an out-sample estimation method, bootstrap tuning, and real-world financial application.

Contribution

It extends multivariate extreme value theory by incorporating directional notions and provides a novel estimation and bootstrap procedure for high-level directional quantiles.

Findings

Asymptotic normality of the estimator is established.

The method performs well on simulated data.

Application to financial data demonstrates practical utility.

Abstract

In multivariate extreme value theory (MEVT), the focus is on analysis outside of the observable sampling zone, which implies that the region of interest is associated to high risk levels. This work provides tools to include directional notions into the MEVT, giving the opportunity to characterize the recently introduced directional multivariate quantiles (DMQ) at high levels. Then, an out-sample estimation method for these quantiles is given. A bootstrap procedure carries out the estimation of the tuning parameter in this multivariate framework and helps with the estimation of the DMQ. Asymptotic normality for the proposed estimator is provided and the methodology is illustrated with simulated data-sets. Finally, a real-life application to a financial case is also performed.