Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth

TL;DR

This paper introduces a new multivariate quantile concept based on directional regression quantiles, establishing their connection with halfspace depth contours and enabling efficient computation and inference in multivariate analysis.

Contribution

The paper proposes a novel multivariate quantile definition using linear programming, linking it with halfspace depth contours and advancing depth-based inference methods.

Findings

Quantiles can be computed efficiently via linear programming.

Contours coincide with Tukey's halfspace depth contours.

Asymptotic properties like Bahadur representation are established.

Abstract

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. Finally, linear programming duality opens the way to promising developments in depth-related multivariate rank-based inference.