\name{rq.fit.lasso}
\alias{rq.fit.lasso}
\title{
Lasso Penalized Quantile Regression
}
\description{
The fitting method implements the lasso penalty of Tibshirani for
fitting quantile regression models. When the argument \code{lambda}
is a scalar the penalty function is the l1
norm of the last (p-1) coefficients, under the presumption that the
first coefficient is an intercept parameter that should not be subject
to the penalty. When \code{lambda} is a vector it should have length
equal the column dimension of the matrix \code{x} and then defines a
coordinatewise specific vector of lasso penalty parameters. In this
case \code{lambda} entries of zero indicate covariates that are not
penalized. There should be a sparse version of this, but isn't (yet).
}
\usage{
rq.fit.lasso(x, y, tau = 0.5, lambda = 1, beta = .99995, eps = 1e-06)
}
\arguments{
\item{x}{
the design matrix
}
\item{y}{
the response variable
}
\item{tau}{
the quantile desired, defaults to 0.5.
}
\item{lambda}{
the value of the penalty parameter(s) that determine how much shrinkage is done.
This should be either a scalar, or a vector of length equal to the column dimension
of the \code{x} matrix.
}
\item{beta}{
step length parameter for Frisch-Newton method.
}
\item{eps}{
tolerance parameter for convergence.
}
}
\value{
Returns a list with a coefficient, residual, tau and lambda components.
When called from \code{"rq"} (as intended) the returned object
has class "lassorqs".
}
\references{
Koenker, R. (2005 \emph{Quantile Regression}, CUP.
}
\author{R. Koenker}
\seealso{
\code{\link{rq}}}
\examples{
n <- 60
p <- 7
rho <- .5
beta <- c(3,1.5,0,2,0,0,0)
R <- matrix(0,p,p)
for(i in 1:p){
for(j in 1:p){
R[i,j] <- rho^abs(i-j)
}
}
set.seed(1234)
x <- matrix(rnorm(n*p),n,p) \%*\% t(chol(R))
y <- x \%*\% beta + rnorm(n)
f <- rq(y ~ x, method="lasso",lambda = 30)
g <- rq(y ~ x, method="lasso",lambda = c(rep(0,4),rep(30,4)))
}
\keyword{regression}