# Lasso Penalized Quantile Regression The fitting method implements the lasso penalty for fitting quantile regression models. When the argument `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 `lambda` is a vector it should have length equal the column dimension of the matrix `x` and then defines a coordinatewise specific vector of lasso penalty parameters. In this case `lambda` entries of zero indicate covariates that are not penalized. If `lambda` is not specified, a default value is selected according to the proposal of Belloni and Chernozhukov (2011). See `LassoLambdaHat` for further details. There should be a sparse version of this, but isn't (yet). There should also be a preprocessing version, but isn't (yet). ```r rq.fit.lasso(x, y, tau = 0.5, lambda = NULL, beta = .99995, eps = 1e-06) ``` ## Arguments - `x`: the design matrix - `y`: the response variable - `tau`: the quantile desired, defaults to 0.5. - `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 `x` matrix. If unspecified, a default value is chosen according to the proposal of Belloni and Chernozhukov (2011). - `beta`: step length parameter for Frisch-Newton method. - `eps`: tolerance parameter for convergence. ## Returns Returns a list with a coefficient, residual, tau and lambda components. When called from `"rq"` (as intended) the returned object has class "lassorqs". ## References Koenker, R. (2005) **Quantile Regression**, CUP. Belloni, A. and V. Chernozhukov. (2011) l1-penalized quantile regression in high-dimensional sparse models. **Annals of Statistics**, 39 82 - 130. ## Author(s) R. Koenker ## See Also `rq` ## Examples ```r 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))) ```

rq.fit.lasso function