The QUANTLIFE Procedure (Experimental)
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 with 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 with 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 with the CI=PW option
in the PROC QUANTLIFE statement.
Consider the linear model
![\[ y_ i =\mb {x}_{1i}^{\prime }\bbeta _1 + \mb {x}_{2i}^{\prime }\bbeta _2 + \epsilon _ i \]](images/statug_quantlife0081.png){kind=small}
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 hypothesis:
![\[ H_{0}: \beta _2(\tau ) = 0 \]](images/statug_quantlife0089.png){kind=small}
The Wald test statistic, which is based on the estimated coefficients 
from the unrestricted fitted model, is given by
![\[ T_ W(\tau ) = {\hat\beta _2^{\prime }(\tau )} {\hat\Sigma (\tau )}^{-1} {\hat\beta _2(\tau )} \]](images/statug_quantlife0091.png){kind=small}
where 
is an estimator of the covariance of 
, which is obtained by using resampling methods.