adaHuber.lasso {adaHuber}

Regularized Adaptive Huber Regression

Description

Sparse regularized Huber regression models in high dimensions with \ell_1 (lasso) penalty. The function implements a localized majorize-minimize algorithm with a gradient-based method.

Usage

adaHuber.lasso(
  X,
  Y,
  lambda = 0.5,
  tau = 0,
  phi0 = 0.01,
  gamma = 1.2,
  epsilon = 0.001,
  iteMax = 500
)

Arguments

X

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

Y

An n-dimensional response vector.

lambda

(optional) Regularization parameter. Must be positive. Default is 0.5.

tau

(optional) The robustness parameter. If not specified or the input value is non-positive, a tuning-free principle is applied. Default is 0 (hence, tuning-free).

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) Tolerance level of the gradient-based algorithm. The iteration will stop when the maximum magnitude of all the elements of the gradient is less than tol. Default is 1e-03.

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.

tau

The robustification parameter calibrated by the tuning-free principle (if the input is non-positive).

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.

See Also

See adaHuber.cv.lasso for regularized adaptive Huber regression with cross-validation.

Examples

n = 200; p = 500; s = 10
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.lasso(X, Y, lambda = 0.5)
beta.lasso = fit.lasso$coef