  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.
