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) defined 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, weighted quantile regression is carried out by solving