From Halfspace M-Depth to Multiple-output Expectile Regression

Despite the importance of expectiles in fields such as econometrics, risk management, and extreme value theory, expectile regression—or, more generally, M-quantile regression—unfortunately remains limited to single-output problems. To improve on this, we define hyperplane-valued multivariate M-quantiles that show strong advantages over their point-valued competitors. Our M-quantiles are directional in nature and provide centrality regions when all directions are considered. These regions define new statistical depths, the halfspace M-depths, that include the celebrated Tukey depth as a particular case. We study thoroughly the proposed M-quantiles, halfspace M-depths, and corresponding regions. M-depths not only provide a general framework to consider Tukey depth, expectile depth, Lr-depths, etc., but are also of interest on their own. However, since our original motivation was to consider multiple-output expectile regression, we pay more attention to the expectile case and show that expectile depth and multivariate expectiles enjoy distinctive properties that will be of primary interest to practitioners: expectile depth is maximized at the mean vector, is smoother than the Tukey depth, and exhibits surprising monotonicity properties that are key for computational purposes. Finally, our multivariate expectiles allow defining multiple-output expectile regression methods, that, in riskoriented applications in particular, are preferable to their analogs based on standard quantiles