On directional multiple-output quantile regression

This paper sheds some new light on the multivariate (projectional) quantiles recently introduced in Kong and Mizera (2008). Contrary to the sophisticated set analysis used there, 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. We also link these quantiles with portfolio optimization and present an alternative proof that the resulting quantile regions coincide with the halfspace depth ones. Our proof makes the link between halfspace depth contours and univariate quantiles of projections more explicit and results into an exact computation of sample quantile regions (hence also of halfspace depth regions) from projectional quantiles. Throughout, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008). Above all, we study the projectional regression quantile regions and compare them with those resulting from the approach considered in Hallin, Paindaveine, and Siman (2009).To gain in generality and to make the comparison between both concepts easier, we present a general framework for directional multivariate(regression) quantiles which includes both approaches as particular cases and is of interest in itself.