Local constant and local bilinear multiple-output quantile regression

A new quantile regression concept, based on a directional version of Koenker and Bassett’s traditional single-output one, has been introduced in [Hallin, Paindaveine and ˇSiman, Annals of Statistics 2010, 635-703] for multiple-output regression problems. The polyhedral contours provided by the empirical counterpart of that concept, however, cannot adapt to nonlinear and/or heteroskedastic dependencies. This paper therefore introduces local constant and local linear versions of those contours, which both allow to asymptotically recover the conditional halfspace depth contours of the response. In the multiple-output context considered, the local linear construction actually is of a bilinear nature. Bahadur representation and asymptotic normality results are established. Illustrations are provided both on simulated and real data.