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 by using resampling methods. The QUANTLIFE procedure implements two different methods, the exponentially weighted method and the pairwise resampling method.

This method samples weights from a standard exponential distribution that has mean 1 and variance 1. Then it computes the censored quantile regression estimators based on the observed data 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 by using the CI=EW option in the PROC QUANTLIFE statement.

This method samples with replacement and computes the quantile regression estimators 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 by using the CI=PW option in the PROC QUANTLIFE statement.

Consider the linear model

where and are p-dimensional and q-dimensional parameters, respectively, and , , are errors. Denote , and let and be the parameter estimates for and , respectively, at the th quantile.

The QUANTLIFE procedure implements the Wald test for the null hypothesis:

The Wald test statistic, which is based on the estimated coefficients from the unrestricted fitted model, is given by

where is an estimator of the covariance of , which is obtained by using resampling methods.