Local bilinear multiple-output quantile/depth regression

TL;DR

This paper introduces local bilinear multiple-output quantile/depth regression methods that adapt to nonlinear and heteroskedastic dependencies, providing asymptotic properties and practical illustrations.

Contribution

It develops local constant and bilinear (local linear) contour estimators for multivariate quantile regression, improving adaptability over previous polyhedral approaches.

Findings

Asymptotic normality of the estimators is established.

The methods effectively recover conditional depth contours.

Illustrations demonstrate practical applicability on simulated and real data.

Abstract

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.