On directional multiple-output quantile regression

This paper sheds some new light on projection quantiles. Contrary to the sophisticated set analysis used in Kong and Mizera (2008) [13], we adopt a more parametric approach and study the subgradient conditions associated with these quantiles. In this setup, we introduce Lagrange multipliers which can be interpreted in various interesting ways, in particular in a portfolio optimization context. The corresponding projection quantile regions were already shown to coincide with the halfspace depth ones in Kong and Mizera (2008) [13], but we provide here an alternative proof (completely based on projection quantiles) that has the advantage of leading to an exact computation of halfspace depth regions from projection quantiles. Above all, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008) [13]. We show in particular that the regression quantile regions introduced in Hallin, Paindaveine, and Šiman (2010) [6] and [7] can also be obtained from projection (regression) quantiles, which may lead to a faster computation of those regions in some particular cases.