An Adapted Loss Function for Censored Quantile Regression

Abstract

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.

Linear censored quantile regression: A novel minimum‐distance approach

“…It is established in this work that the latter class is in some sense well behaved in terms of size, which is represented by the notion of bracketing numbers (see Van der Vaart and Wellner, 1996, p. 83). Note that this condition is somewhat similar to condition (C3) in De Backer et al (2019), but extended here for the inclusion of the dimension reduction framework. Conditions (C6) and (C7) are likewise classical in the literature on censored quantile regression, and may for instance be also found in the work of De Backer et al (2019) and Wang and Wang (2009) to name a few.…”

“…Note that this condition is somewhat similar to condition (C3) in De Backer et al (2019), but extended here for the inclusion of the dimension reduction framework. Conditions (C6) and (C7) are likewise classical in the literature on censored quantile regression, and may for instance be also found in the work of De Backer et al (2019) and Wang and Wang (2009) to name a few. In addition, Assumption (C7) is to be brought in parallel with previously developed comments on constraints on the quantile level when confronted to censored data, as the latter condition is observed to define a natural upper bound on the quantile level one may consider in this context.…”

“…This is due to the fact that the asymptotic covariance matrix depends on several unknown quantities that are cumbersome to estimate in practice. Similarly to Portnoy (2003), Wang and Wang (2009), and De Backer et al (2019) to name a few, we therefore consider here for inference a simple percentile bootstrap procedure, where 95% bootstrap confidence intervals for are constructed by taking the 2.5th and 97.5th percentiles of the bootstrap coefficients, obtained by drawing a sufficient amount of bootstrap samples through resampling with replacement. The validity of the latter technique in our framework will be empirically investigated in Section 4.3.…”

“…As a result, the literature based on this approach is noticeably more modest. The second attempt of changing the “target” was recently proposed in De Backer, El Ghouch, and Van Keilegom (2019), where it is suggested to work on the loss function itself instead of considering weighting schemes or alternative quantile levels. The methodology results in a simple adaptation of the check function has a wide application range in terms of modeling potential and is illustrated for the particular case of linear regression.…”

Multistate quantile regression models

“…There are proposals to estimate censored QR parameters using a modified Kaplan‐Meier estimator, martingale‐based estimating equations, as well as weighted estimating equations . Recently, data augmentation and a modification of the check loss function for censoring have been proposed. More in general, time‐to‐event data comprise possibly complex settings, such as recurrent events, competing risks, and semicompeting risks.…”