The QUANTSELECT Procedure

Quantile Regression for Extremal Quantile Levels

A quantile level tau is extremal if tau 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 tau element-of left-bracket StartFraction 1 Over 4 n EndFraction comma 1 minus StartFraction 1 Over 4 n EndFraction right-bracket, tau is not extremal. Otherwise, follow these steps:

  1. Set tau 0 equals StartFraction 1 Over 4 n EndFraction left-parenthesis or tau 0 equals left-parenthesis 1 minus StartFraction 1 Over 4 n EndFraction right-parenthesis right-parenthesis.

  2. Compute ModifyingAbove bold-italic beta With caret left-parenthesis tau 0 right-parenthesis equals normal a normal r normal g normal m normal i normal n Underscript bold-italic beta Endscripts sigma-summation Underscript i equals 1 Overscript n Endscripts rho Subscript tau 0 Baseline left-parenthesis y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta right-parenthesis.

  3. Find the quantile-level lower limit (or upper limit), tau 1, such that ModifyingAbove bold-italic beta With caret left-parenthesis tau 0 right-parenthesis is still optimal at tau 1.

  4. If tau 1 less-than-or-equal-to tau (or tau 1 greater-than-or-equal-to tau), return ModifyingAbove bold-italic beta With caret left-parenthesis tau 0 right-parenthesis. Otherwise, update tau 0 equals tau 1 minus c (or tau 0 equals tau 1 plus c) for a small tolerance c greater-than 0, and go to step 2.

Last updated: December 09, 2022