# The QUANTSELECT Procedure

#### Quantile Regression for Extremal Quantile Levels

A quantile level is extremal if is equal to or approaching 0 or 1. The solution for an extremal quantile-level quantile regression problem can be nonunique because the parameter estimate of the intercept effect can be arbitrarily small or large. In a quantile process regression toward the direction of the specified extremal quantile level, the tightest solution refers to the first solution whose quantile-level range includes the specified extremal quantile level. Among all the valid solutions for an extremal quantile-level quantile regression problem, the tightest solution can generalize the terminology of sample minimum and sample maximum.

The QUANTSELECT procedure computes the tightest solution for an extremal quantile-level quantile regression problem by using the ALGORITHM=SIMPLEX algorithm. If , is not extremal. Otherwise, follow these steps:

1. Set .

2. Compute .

3. Find the quantile-level lower limit (or upper limit), , such that is still optimal at .

4. If (or ), return . Otherwise, update (or ) for a small tolerance , and go to step 2.