Linear censored quantile regression: A novel minimum‐distance approach

Abstract

In this article, we investigate a new procedure for the estimation of a linear quantile regression with possibly right-censored responses. Contrary to the main literature on the subject, we propose in this context to circumvent the formulation of conditional quantiles through the so-called "check" loss function that stems from the influential work of Koenker and Bassett (1978). Instead, our suggestion is here to estimate the quantile coefficients by minimizing an alternative measure of distance. In fact, our approach could be qualified as a generalization in a parametric regression framework of the technique consisting in inverting the conditional distribution of the response given the covariates. This is motivated by the knowledge that the main literature for censored data already relies on some nonparametric conditional distribution estimation as well. The ideas of effective dimension reduction are then exploited in order to accommodate for higher dimensional settings as well in this context. Extensive numerical results then suggest that such an approach provides a strongly competitive procedure to the classical approaches based on the check function, in fact both for complete and censored observations. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity

The comparison of censored quantile regression methods in prognosis factors of breast cancer survival

“…De backer et al in their study indicated in an extensive simulation study that the resulted quantile regression estimator respect to established check-based formulations have less variance results. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity conditions 10 . Yang et al indicated that the Yang's method presents an estimator is able to achieve significant efficiency gains in comparisons with Portnoy’s estimator 9 .…”

“…The most applying methods during the recent years are Portnoy 5 , Wang and Wang 7 , Bottai and Zhang 8 , Yang et al 9 and De Backer et al 10 methods. In the following, we present a brief overview of methodological framework for these models.…”

“…De Backer et al 10 proposed to estimate model ( 4 ) based on a minimum distance loss function, given by . They further suggested using a smooth double kernel version of .…”

“…For conditionally random right censoring, various approaches have been proposed that the most practical of them include: Portnoy 5 , Peng and Huang 6 , Wang and Wang 7 , Bottai and Zhang 8 , Yang et al 9 and De Backer et al 10 methods. Portnoy generalized the Kaplan–Meier method to estimate quantile of survival time with a recursively weighted estimation algorithm under the global linearity assumption of the conditional quantile functions 5 .…”

“…Base on the general principle of data augmentation 15 , in the algorithm, they first, impute censored values from the quantile functions then using the imputed values fit the quantile model. De Backer et al investigate a new procedure for modeling right-censored data with a linear quantile regression 10 . They used “check” loss function that stems from the influential work of Koenker and Bassett 16 , to circumvent the formulation of conditional quantiles.…”