Computing multiple-output regression quantile regions from projection quantiles

In the multiple-output regression context, Hallin et al. (Ann Statist 38:635-669, 2010) introduced a powerful data-analytical tool based on regression quantile regions. However, the computation of these regions, that are obtained by considering in all directions an original concept of directional regression quantiles, is a very challenging problem. Paindaveine and iman (Comput Stat Data Anal 2011b) described a first elegant solution relying on linear programming techniques. The present paper provides another solution based on the fact that the quantile regions can also be computed from a competing concept of projection regression quantiles, elaborated in Kong and Mizera (Quantile tomography: using quantiles with multivariate data 2008) and Paindaveine and iman (J Multivar Anal 2011a). As a by-product, this alternative solution further provides various characteristics useful for statistical inference. We describe in detail the algorithm solving the parametric programming problem involved, and illustrate the resulting procedure on simulated data. We show through simulations that the Matlab implementation of the algorithm proposed in this paper is faster than that from Paindaveine and iman (Comput Stat Data Anal 2011b) in various cases.