  Despite the renewed interest in the Newey and Powell (1987) concept 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 introduce hyperplane-valued multivariate M-quantiles that
show strong advantages, for instance in terms of equivariance, over the various
point-valued multivariate M-quantiles available in the literature. Like their
competitors, our multivariate M-quantiles are directional in nature and provide
centrality regions when all directions are considered. These regions define a
new statistical depth, the halfspace M-depth, whose deepest point, in the
expectile case, is the mean vector. Remarkably, the halfspace M-depth can
alternatively be obtained by substituting, in the celebrated Tukey (1975)
halfspace depth, M-quantile outlyingness for standard quantile outlyingness,
which supports a posteriori the claim that our multivariate M-quantile concept
is the natural one. We investigate thoroughly the properties of the proposed
multivariate M-quantiles, of halfspace M-depth, and of the corresponding
regions. Since our original motivation was to define multiple-output expectile
regression methods, we further focus on the expectile case. We show in
particular that expectile depth is smoother than the Tukey depth and enjoys
interesting monotonicity properties that are extremely promising for
computational purposes. Unlike their quantile analogs, the proposed
multivariate expectiles also satisfy the coherency axioms of multivariate risk
measures. Finally, we show that our multivariate expectiles indeed allow
performing multiple-output expectile regression, which is illustrated on
simulated and real data.
