R: Lambda selection for QR lasso problems

LassoLambdaHat {quantreg} | R Documentation

## 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)

### 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`.

### Value

vector of default lambda values of length p, the column dimension of X.

### References

Belloni, A. and V. Chernozhukov. (2011) l1-penalized quantile regression in high-dimensional sparse models. Annals of Statistics, 39 82 - 130.

### Examples

    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

[Package quantreg version 5.93 Index]