Multivariate Expectiles, Expectile Depth and Multiple-Output Expectile Regression

Despite the importance of expectiles in fields such as econometrics, risk management, and extreme value theory, expectile regression unfortunately so far remains limited to single-output problems. To improve on this, we define hyperplane-valued multivariate expectiles that show strong advantages over their point-valued competitors. Our expectiles are directional in nature and provide centrality regions when all directions are considered. These regions define a new statistical depth, the halfspace expectile depth, that is an L2 version of the celebrated (L1) Tukey halfspace depth. We study thoroughly the proposed expectiles, the expectile depth, and the corresponding regions. When compared to their L1 counterparts, these concepts enjoy distinctive properties that will be of primary interest to practitioners. In particular, expectile depth is maximized at the mean vector, is smoother than the halfspace depth, and exhibits surprising monotonicity properties that are key for computational purposes. Finally, the proposed multivariate expectiles allow us to define multiple-output ex- pectile regression methods, that, in risk-oriented applications in particular, dominate their analogs based on standard quantiles.