Statistical Inference in Quantile Regression Models

The main purpose of this dissertation is to collect different innovative statistical methods in quantile regression. The contributions can be summarized as follows: -- A new method to construct prediction intervals involving median regression and bootstrapping the prediction error is proposed. -- A plug-in bandwidth selector for nonparametric quantile regression has been proposed, that is based on nonparametric estimations of the curvature of the quantile regression function and the integrated sparsity. -- Two lack-of-fit tests for quantile regression models have been presented. The first test is based on the cumulative sum of residuals with respect to unidimensional linear projections of the covariates in order to deal with high-dimensional covariates. The second test is based on interpreting the residuals from the quantile model fit as response values of a logistic regression. Then a likelihood ratio test in the logistic regression is used to check the quantile model