scad: The SCAD 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 SCAD penalty (Fan \& Li, 2001)

1	scad(lambda = NULL, ...)

`lambda`	two-dimensional tuning parameter. The first component corresponds to the regularization parameter λ that drives the relevance of the SCAD penalty for likelihood inference. It must be nonnegative. The second component corresponds to a (see details below). It must be greater than two.
`...`	further arguments.

The SCAD penalty is formally defined as

P_{\tilde{λ}}^{sc} (\boldsymbol{β}) = ∑_{j=1}^p p_{\tilde{λ},j}^{sc} (|β_j|), \quad \tilde{λ} = (λ, s),

where p_{\tilde{λ},j}^{sc} (|β_j|) is complicated to be specified directly. Fan \& Li (2001) just give the penalty by the first derivatives of its components as

\frac{d p_{\tilde{λ},j}^{sc} (|β_j|)}{d |β_j|} = λ≤ft\{1_{|β_j| ≤q λ}(|β_j|) + \frac{(aλ - |β_j|)_+}{(a-1)λ}1_{|β_j| > λ} (|β_j|) \right\},

where we use the notation b_+ := \max \{0, b\} and 1_A(x) denotes the indicator function. The penalty depends on two tuning parameters, λ>0 and a>2. It is continuously differentiable in β_j, but not in their tuning parameters. If |β_j| ≤q λ then the lasso penalty is applied to β_j. Afterwards this penalization smoothly clipped apart until the threshold a is reached. For |β_j| > a there is no penalization at all at this coefficient. Fan \& Li (2001) suggest to use a = 3.7. The SCAD penalty leaves large values of β_j not excessively penalized and makes the solution continuous.

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

`penalty`	character: the penalty name.
`lambda`	double: the (nonnegative) tuning parameter.
`getpenmat`	function: computes the diagonal penalty matrix.

Jan Ulbricht

Fan, J. \& R. Li (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360.

penalty, lasso, ridge

lqa documentation built on May 30, 2017, 3:41 a.m.