Davy Paindaveine (Université Libre de Bruxelles-ECARES), “Multivariate quantiles and multiple-output regression quantiles: From L_1 optimization to halfspace depth ”, Statistics Seminar, Toulouse: TSE, May 10, 2011, 14:00–15:30, room MF 323.
A new multivariate concept of quantile, based on a directional version of Koenker and Bassett's traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. In its original version, this new regression quantile concept produces, in the empirical case, polyhedral contours that cannot adapt to nonlinear or/and heteroskedastic dependencies. We therefore further propose a local version of those contours, which asymptotically recovers the conditional halfspace depth contours of the multiple-output response. Examples are provided.