adaHuber.cv.lasso: Cross-Validated Regularized Adaptive Huber Regression.

Description

Sparse regularized adaptive Huber regressionwith "lasso" penalty. The function implements a localized majorize-minimize algorithm with a gradient-based method. The regularization parameter \(\lambda\) is selected by cross-validation, and the robustification parameter \(\tau\) is determined by a tuning-free principle.

Usage

adaHuber.cv.lasso(
  X,
  Y,
  lambdaSeq = NULL,
  kfolds = 5,
  numLambda = 50,
  phi0 = 0.01,
  gamma = 1.2,
  epsilon = 0.001,
  iteMax = 500
)

Arguments

A \(n\) by \(p\) design matrix. Each row is a vector of observation with \(p\) covariates.

An \(n\)-dimensional response vector.

lambdaSeq

(optional) A sequence of candidate regularization parameters. If unspecified, a reasonable sequence will be generated.

kfolds

(optional) Number of folds for cross-validation. Default is 5.

numLambda

(optional) Number of \(\lambda\) values for cross-validation if lambdaSeq is unspeficied. Default is 50.

phi0

(optional) The initial quadratic coefficient parameter in the local adaptive majorize-minimize algorithm. Default is 0.01.

gamma

(optional) The adaptive search parameter (greater than 1) in the local adaptive majorize-minimize algorithm. Default is 1.2.

epsilon

(optional) A tolerance level for the stopping rule. The iteration will stop when the maximum magnitude of the change of coefficient updates is less than epsilon. Default is 0.001.

iteMax

(optional) Maximum number of iterations. Default is 500.

Value

An object containing the following items will be returned:

coef: A \((p + 1)\) vector of estimated sparse regression coefficients, including the intercept.
lambdaSeq: The sequence of candidate regularization parameters.
lambda: Regularization parameter selected by cross-validation.
tau: The robustification parameter calibrated by the tuning-free principle.
iteration: Number of iterations until convergence.
phi: The quadratic coefficient parameter in the local adaptive majorize-minimize algorithm.

References

Pan, X., Sun, Q. and Zhou, W.-X. (2021). Iteratively reweighted l1-penalized robust regression. Electron. J. Stat., 15, 3287-3348.

Sun, Q., Zhou, W.-X. and Fan, J. (2020). Adaptive Huber regression. J. Amer. Statist. Assoc., 115 254-265.

Wang, L., Zheng, C., Zhou, W. and Zhou, W.-X. (2021). A new principle for tuning-free Huber regression. Stat. Sinica, 31, 2153-2177.

Examples

Run this code

# NOT RUN {
n = 100; p = 200; s = 5
beta = c(rep(1.5, s + 1), rep(0, p - s))
X = matrix(rnorm(n * p), n, p)
err = rt(n, 2)
Y = cbind(rep(1, n), X) %*% beta + err 

fit.lasso = adaHuber.cv.lasso(X, Y)
beta.lasso = fit.lasso$coef
# }