  A new quantile regression concept, based on a directional version of Koenker
and Bassett's traditional single-output one, has been introduced in [Ann.
Statist. (2010) 38 635-669] for multiple-output location/linear regression
problems. The polyhedral contours provided by the empirical counterpart of that
concept, however, cannot adapt to unknown nonlinear and/or heteroskedastic
dependencies. This paper therefore introduces local constant and local linear
(actually, bilinear) versions of those contours, which both allow to
asymptotically recover the conditional halfspace depth contours that completely
characterize the response's conditional distributions. Bahadur representation
and asymptotic normality results are established. Illustrations are provided
both on simulated and real data.
