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 $\displaystyle \tau \in \left[{1\over 4n},1-{1\over 4n}\right]$, $\tau $ is not extremal. Otherwise, follow these steps:

  1. Set $\displaystyle \tau _0={1\over 4n}$ $\left(\mbox{ or } \displaystyle \tau _0=\left(1-{1\over 4n}\right)\right)$.

  2. Compute $\displaystyle \hat{\bbeta }(\tau _0)=\underset {\bbeta }{\mr{arg min}} \sum _{i=1}^ n \rho _{\tau _0} \left(y_ i-\mb{x}_ i^{\prime }\bbeta \right)$.

  3. Find the quantile-level lower limit (or upper limit), $\tau _1$, such that $\hat{\bbeta }(\tau _{0})$ is still optimal at $\tau _1$.

  4. If $\tau _1\le \tau $ (or $\tau _1\ge \tau $), return $\hat{\bbeta }(\tau _0)$. Otherwise, update $\tau _0=\tau _1-c$ (or $\tau _0=\tau _1+c$) for a small tolerance $c>0$, and go to step 2.