# Bridge Penalty

### Description

Object of the `penalty`

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

### Usage

1 |

### Arguments

`lambda` |
two dimensional tuning parameter parameter. The first component corresponds to the regularization parameter |

`...` |
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. |

### Author(s)

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.

### See Also

`penalty`

, `lasso`

, `ridge`

, `ao`

, `genet`