  In this paper, 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 paper 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 modelling techniques.
In order 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 of the estimator for linear regression is obtained using
very recent results on non-smooth 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.
