Quantile regression with censored data : an investigation of new estimation procedures

One of the most fundamental purposes of applied statistics is to provide an appropriate description of the relationship between certain covariate variables and a response of interest. While conditional mean models historically dominated the regression landscape for this task, the last decades have witnessed the emergence of a variety of regression models focusing on conditional quantiles instead, as the latter allow, among other conveniences, for a more complete representation of the relationship of interest. Providing convenient tools for estimating a quantile regression model then constitutes a major challenge for statisticians. This statement is further emphasized in situations where the data at hand slightly diverges from the ideal scenario where all observations are completely recorded. One example of such nuisance in the data often encountered in practice concerns the occurrence of right-censoring of the responses, where one no longer completely observes the true response of interest but instead only the minimum of it and an interfering censoring variable. In this context, this thesis aims at developing new estimation procedures for quantile regression models where right-censoring of the response may occur, although part of our work encompasses some novelties for strictly complete observations as well. Our exploration will lead us from the exploitation of copulas in regression modelling, to the consideration of adapting the loss function usually employed for quantiles in order to take censoring into account, before examining at last the formulation of a new minimum-distance approach in this context.