adaptive.lasso: Adaptive Lasso Penalty

Description

Object of the penalty class to handle the adaptive lasso penalty (Zou, 2006).

Usage

adaptive.lasso (lambda = NULL, al.weights = NULL, ...)

Arguments

lambda

regularization parameter. This must be a nonnegative real number.

al.weights

weights used for the adaptive lasso penalty.

...

further arguments.

Value

An object of the class penalty. This is a list with elements
penaltycharacter: the penalty name.
lambdadouble: the (nonnegative) regularization parameter.
getpenmatfunction: computes the diagonal penalty matrix.
first.derivativefunction: This returns the p-dimensional vector of the first derivative of the p penalty terms with respect to $|\beta_i|$. However, this function is not really required for adaptive lasso.

Details

The adaptive lasso penalty (Zou, 2006) is defined as $$P_{\tilde{\lambda}}^{al} (\boldsymbol{\beta}) = \lambda \sum_{i=1}^p w_i |\beta_i|,$$ where adaptive weights $w_i$ are used for penalizing different coefficients in the $L_1$-norm penalty. Based on a root-n-consistent estimator $\hat{\boldsymbol{\beta}}$ of the true parameter vector $\boldsymbol{\beta}$, Zou (2006) uses the weight vector $\hat\mathbf{w} = (\hat{w}_1,\ldots,\hat{w}_p)^\top$ with $$\hat{w}_j = \frac{1}{|\hat\beta_j|^\gamma}, \quad j = 1, \ldots, p$$ as estimates for the adaptive weights, where $\gamma > 0$ can be chosen arbitrarily. In the $n > p$ case, the MLE can be used for the estimated weights. In the $n \ll p$ case, a ridge penalized MLE with an optimized regularization parameter might be a good alternative. You can commit any nonnegative weights by using al.weights in the initialization of the corresponding penalty object. If you left it unspecified then al.weights = 1 will be used.

References