An Adapted Loss Function for Censored Quantile Regression

ABSTRACT

In this article, we study a novel approach for the estimation of quantiles when facing potential right censoring of the responses. Contrary to the existing literature on the subject, the adopted strategy of this article is to tackle censoring at the very level of the loss function usually employed for the computation of quantiles, the so-called “check” function. For interpretation purposes, a simple comparison with the latter reveals how censoring is accounted for in the newly proposed loss function. Subsequently, when considering the inclusion of covariates for conditional quantile estimation, by defining a new general loss function the proposed methodology opens the gate to numerous parametric, semiparametric, and nonparametric modeling techniques. To illustrate this statement, we consider the well-studied linear regression under the usual assumption of conditional independence between the true response and the censoring variable. For practical minimization of the studied loss function, we also provide a simple algorithmic procedure shown to yield satisfactory results for the proposed estimator with respect to the existing literature in an extensive simulation study. From a more theoretical prospect, consistency and asymptotic normality of the estimator for linear regression are obtained using several recent results on nonsmooth semiparametric estimation equations with an infinite-dimensional nuisance parameter, while numerical examples illustrate the adequateness of a simple bootstrap procedure for inferential purposes. Lastly, an application to a real dataset is used to further illustrate the validity and finite sample performance of the proposed estimator. Supplementary materials for this article are available online.

Keywords:

• Beran estimator
• Bootstrap
• Check function
• Linear regression
• MM algorithm

Supplementary Material

Proofs: Technical proofs of Section 3. (pdf file)

Acknowledgments

All the authors thank two anonymous referees, the associate editor, and the editor whose comments have helped to improve the article substantially.

Additional information

Funding

All authors acknowledge financial support from IAP research network P7/06 of the Belgian Government (Belgian Science Policy). M. De Backer and A. El Ghouch further acknowledge financial support from the FSR project IMAQFSR15PROJEL from the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS). I. Van Keilegom also acknowledges support from the European Research Council (2016-2021, Horizon 2020 / ERC grant agreement No. 694409).Computational resources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI) funded by the Fonds de la Recherche Scientifique de Belgique under convention 2.5020.11.