quantreg (version 5.98)

rq.fit.scad: SCADPenalized Quantile Regression

Description

The fitting method implements the smoothly clipped absolute deviation penalty of Fan and Li for fitting quantile regression models. When the argument lambda is a scalar the penalty function is the scad modified 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 scad penalty parameters. In this case 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.scad(x, y, tau = 0.5, alpha = 3.2, lambda = 1, start="rq", 
	beta = .9995, eps = 1e-06)

Value

Returns a list with a coefficient, residual, tau and lambda components. When called from "rq" as intended the returned object has class "scadrqs".

Arguments

x

the design matrix

y

the response variable

tau

the quantile desired, defaults to 0.5.

alpha

tuning parameter of the scad penalty.

lambda

the value of the penalty parameter that determines 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.

start

starting method, default method 'rq' uses the unconstrained rq estimate, while method 'lasso' uses the corresponding lasso estimate with the specified lambda.

beta

step length parameter for Frisch-Newton method.

eps

tolerance parameter for convergence.

Author

R. Koenker

Details

The algorithm is an adaptation of the "difference convex algorithm" described in Wu and Liu (2008). It solves a sequence of (convex) QR problems to approximate solutions of the (non-convex) scad problem.

References

Wu, Y. and Y. Liu (2008) Variable Selection in Quantile Regression, Statistica Sinica, to appear.

See Also

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="scad",lambda = 30)
g <- rq(y ~ x, method="scad", start = "lasso", lambda = 30)