\name{WRegEst}
\alias{WRegEst}
\title{Compute the casewise weighted regression estimator for AFT model}
\usage{
WRegEst(x, y, delta, LS=TRUE, tau=0.5)
}
\arguments{
    \item{x}{a matrix of size N by q. }
    \item{y}{a vector of length N, containing the censored responses. Usually 
             the log of the original observed failure times. }
    \item{delta}{a vector (length N) of either 1's or 0's. 
              d=1 means y is uncensored;
	      d=0 means y is right censored. }
    \item{LS}{a logical value. If TRUE then the function will 
              return the least squares estimator. If FALSE then the
              function will return the regression quantile estimator,
              with quantile specified by tau. }.
    \item{tau}{a scalar, between 0 and 1. The quantile to be used in
                 quantile regression. If LS=TRUE, then it is ignored.}
}
\description{
For the AFT model, this function computes the case weighted estimator of
beta. Either the least squares estimator or the regression quantile estimator.
}
\details{
Due to the readily available minimizer, we only provide least squares
and quantile regression here. However, in the companion testing function
\code{WRegTest } the user can supply a self defined psi function,
corresponding to the general M-estimation in the regression modeling.
(since there is no minimization needed).

The estimator is the minimizer of
\deqn{ \sum_{i=1}^n w_i \rho (Y_i - X_i b) }
Assuming a correlation model
\deqn{ Y_i = X_i \beta + \sigma(X_i) \epsilon_i },
where \eqn{\rho( )} is either the square or the absolute value function.
}
\value{
The estimator \eqn{ \hat \beta}.
}
\references{
    Zhou, M.; Bathke, A. and Kim, M. (2006). 
       Empirical likelihood analysis of the
       Heteroscastic AFT model. Dept. of Statistics, Univ. of Kentucky
       Tech. Report. 
}
\author{ Mai Zhou. }
\examples{
data(smallcell)
WRegEst(x=cbind(1,smallcell[,1],smallcell[,2]), 
        y=smallcell[,3], delta=smallcell[,4]) 
####################################################
#### you should get         x1         x2         x3
####                 -59.22126 -488.41306   16.03259
####################################################
WRegEst(x=cbind(1,smallcell[,1],smallcell[,2]), 
        y=log10(smallcell[,3]), delta=smallcell[,4], LS=FALSE) 
########################################################
#### you should get      
####     [1]     2.603342985  -0.263000044  0.003836832
########################################################
xx <- c(28,-44,29,30,26,27,22,23,33,16,24,29,24,40,21,31,34,-2,25,19)
}
\keyword{nonparametric}