On the estimation of extreme directional multivariate quantiles

In multivariate extreme value theory, the focus is on the analysis outside of the observable sampling zone, which implies that a region of interest is associated to high risk levels. This work provides an out-sample estimation method for the recently introduced directional multivariate quantiles (DMQ) and a characterization of these quantiles at high levels. We introduce a multivariate procedure to carry out the estimation of the DMQ and the asymptotic normality of the proposed estimator is derived. The methodology is illustrated with a multivariate t-distribution for which the theoretical DMQ are analytically known. A real-life application to a financial case is also performed.