This section describes the basic concepts and notations for quantile regression and quantile regression model selection.

Let denote a data set of observations, where are responses and are regressors. Koenker and Bassett (1978) define the *regression quantile* at quantile level as any solution to the minimization problem

where is a check loss function in which and .

If you specify weights , in the WEIGHT statement, then weighted quantile regression is carried out by solving

The HPQUANTSELECT procedure fits a quantile regression model by using a predictor-corrector interior point algorithm, which was originally designed to solve support vector machine classifiers for large data sets (Gertz and Griffin, 2005, 2010).