Confidence Interval :: SAS/STAT(R) 12.1 User's Guide

Confidence Interval

Exponentially Weighted Method
Pairwise Method
Testing Effects of Covariates

Direct computation of the covariance of the parameter estimators involves a complicated density estimation. Instead, the QUANTLIFE procedure computes confidence intervals for the quantile regression parameters ${\beta }(\tau )$ by using resampling methods. The QUANTLIFE procedure implements two different methods, the exponentially weighted method and the pairwise resampling method.

Exponentially Weighted Method

This method samples weights from a standard exponential distribution with mean 1 and variance 1. Then it computes the censored quantile regression estimators $\hat\beta (\tau )$ based on the observed data $(x_ i,Y_ i, \Delta _ i)$ with the weights . These steps are repeated B times (where B is the value of the NREP= option in the PROC QUANTLIFE statement). The confidence intervals can be obtained from these B estimates. You can specify this method with the CI=EW option in the PROC QUANTLIFE statement.

Pairwise Method

This method samples $(x_ i,Y_ i, \Delta _ i)$ with replacement and computes the quantile regression estimators $\hat\beta (\tau )$ based on the resampled data. These steps are repeated B times (where B is the value of the NREP= option in the PROC QUANTLIFE statement). The confidence intervals can be obtained from these B estimates. You can specify this method with the CI=PW option in the PROC QUANTLIFE statement.

Testing Effects of Covariates

Consider the linear model

$y_ i =\mb {x}_{1i}^{\prime }\bbeta _1 + \mb {x}_{2i}^{\prime }\bbeta _2 + \epsilon _ i$

where $\bbeta _1$ and $\bbeta _2$ are p-dimensional and q-dimensional parameters, respectively, and $\epsilon _ i$ , $i=1,\ldots ,n$ , are errors. Denote $\mb {x}_ i^{\prime }=(\mb {x}_{1i}^{\prime }, \mb {x}_{2i}^{\prime })$ , and let $\hat\bbeta _1(\tau )$ and $\hat\bbeta _2(\tau )$ be the parameter estimates for $\bbeta _1$ and $\bbeta _2$ , respectively, at the $\tau$ th quantile.