ABSTRACT
All multiple-output quantile regression concepts presented so far were based on directional extensions of the usual single-output ones. Direct approaches are possible, though, along three main lines. The first is based on a spatial extension of the definition of the check function, leading to "spatial" quantiles. The second involves substituting, in the traditional Koenker and Bassett definition, ellipsoids for hyperplanes, and the "above/below" indicators with "outside/inside" ones. The third approach, inspired by the relation between directional quantiles and halfspace depth is based on recent measure transportation-related concepts of Monge–Kantorovich depth and quantiles. The concepts of spatial median and quantiles are based on an alternative form for the check function that "naturally" extends to a multivariate context, possibly combined with transformation–retransformation ideas. The spatial regression quantiles partially order the regression hyperplanes.
