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 that minimizes the following objective function in
:
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