# bridge: Bridge Penalty

### Description

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

### Usage

 1 bridge (lambda = NULL, ...) 

### Arguments

 lambda two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter λ. This must be a nonnegative real number. The second component indicates the exponent γ of the penalty term. It must hold that γ > 1. ... further arguments.

### Details

The bridge penalty has been introduced in Frank \& Friedman (1993). See also Fu (1998). It is defined as

P_{\tilde{λ}}^{br} (\boldsymbol{β}) = λ ∑_{i=1}^p |β_i|^γ, \quad γ > 0,

where \tilde{λ} = (λ, γ). It features an additional tuning parameter γ 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 (γ = 1) and the ridge penalty (γ = 2) as special cases.

### Value

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

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

Jan Ulbricht

### 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.

penalty, lasso, ridge, ao, genet