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:
Set
.
Compute .
Find the quantile-level lower limit (or upper limit), , such that
is still optimal at
.
If (or
), return
. Otherwise, update
(or
) for a small tolerance
, and go to step 2.