Quantile regression: estimation and lack-of-fit tests

Although mean regression achieved its greatest diffusion in the twentieth century, it is very surprising to observe that the ideas of quantile regression appeared earlier. While the beginning of the least-squares regression can be dated in the year 1805 by the work of Legendre, in the mid-eighteenth century Boscovich already adjusted data on the ellipticity of the Earth using concepts of quantile regression. Quantile regression is employed when the aim of the study is centred on the estimation of the different positions (quantiles). This kind of regression allows a more detailed description of the behaviour of the response variable, adapts to situations under more general conditions of the error distribution and enjoys robustness properties. For all that, quantile regression is a very useful statistical technology for a large diversity of disciplines. In this paper a review on quantile regression methods will be presented