LassoLambdaHat: Lambda selection for QR lasso problems
Description
Default procedure for selection of lambda in lasso constrained
quantile regression as proposed by Belloni and Chernozhukov (2011)
Usage
LassoLambdaHat(X, R = 1000, tau = 0.5, C = 1, alpha = 0.95)
Value
vector of default lambda values of length p, the column dimension of X.
Arguments
X
Design matrix
R
Number of replications
tau
quantile of interest
C
Cosmological constant
alpha
Interval threshold
Details
As proposed by Belloni and Chernozhukov, a reasonable default lambda
would be the upper quantile of the simulated values. The procedure is based
on idea that a simulated gradient can be used as a pivotal statistic.
Elements of the default vector are standardized by the respective standard deviations
of the covariates. Note that the sqrt(tau(1-tau)) factor cancels in their (2.4) (2.6).
In this formulation even the intercept is penalized. If the lower limit of the
simulated interval is desired one can specify alpha = 0.05.
References
Belloni, A. and V. Chernozhukov. (2011) l1-penalized quantile regression
in high-dimensional sparse models. Annals of Statistics, 39 82 - 130.
n <- 200
p <- 10
x <- matrix(rnorm(n*p), n, p)
b <- c(1,1, rep(0, p-2))
y <- x %*% b + rnorm(n)
f <- rq(y ~ x, tau = 0.8, method = "lasso")
# See f$lambda to see the default lambda selection