bridge: Bridge Penalty

Description

Object of the penalty to handle the bridge penalty (Frank & Friedman, 1993, Fu, 1998)

Usage

bridge (lambda = NULL, ...)

Arguments

lambda

two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter $\lambda$. This must be a nonnegative real number. The second component indicates the exponent $\gamma$ of the penalty term. It must hold that $\g

...

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.

Details

The bridge penalty has been introduced in Frank & Friedman (1993). See also Fu (1998). It is defined as $$P_{\tilde{\lambda}}^{br} (\boldsymbol{\beta}) = \lambda \sum_{i=1}^p |\beta_i|^\gamma, \quad \gamma > 0,$$ where $\tilde{\lambda} = (\lambda, \gamma)$. It features an additional tuning parameter $\gamma$ that controls the degree of preference for the estimated coefficient vector to align with the original, hence standardized, data axis directions in the regressor space. It comprises the lasso penalty ($\gamma = 1$) and the ridge penalty ($\gamma = 2$) as special cases.

References

Frank, I. E. & J. H. Friedman (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109--148.

Fu, W. J. (1998) Penalized Regression: the bridge versus the lasso. Journal of Computational and Graphical Statistics 7, 397--416.