licb: L1-Norm based Improved Correlation-based Penalty
In lqa: Penalized Likelihood Inference for GLMs

Description Usage Arguments Details Value Author(s) References See Also

Object of the penalty class to handle the L1-Norm based Improved Correlation-Based (LICB) Penalty (Ulbricht, 2010).

1	licb (lambda = NULL, ...)

`lambda`	two-dimensional tuning parameter parameter. The first component corresponds to the regularization parameter λ_1 for the lasso penalty term, the second one λ_2 for the L_1-norm based correlation penalty term. Both parameters must be nonnegative.
`...`	further arguments

The improved correlation-based (LICB) penalty is defined as

P_{λ}^{licb}(\boldsymbol{β}) = λ_1 ∑_{i=1}^p |β_i| + λ_2 ∑_{i=1}^{p-1} ∑_{j > i} ≤ft\{\frac{|β_i - β_j|}{1 - \varrho_{ij}} + \frac{|β_i + β_j|}{1 + \varrho_{ij}}\right\}.

The LICB has been introduced to overcome the major drawback of the correlation based-penalized estimator, that is its lack of sparsity. See Ulbricht (2010) for details.

An object of the class penalty. This is a list with elements

`penalty`	character: the penalty name.
`lambda`	double: the (nonnegative) regularization parameter.
`first.derivative`	function: This returns the J-dimensional vector of the first derivative of the J penalty terms with respect to \|\mathbf{a}^\top_j\boldsymbol{β\|}.
`a.coefs`	function: This returns the p-dimensional coefficient vector \mathbf{a}_j of the J penalty terms.