# penalty: penalty In LINselect: Selection of Linear Estimators

## Description

Calculate the penalty function for estimators selection.

## Usage

 `1` ```penalty(Delta, n, p, K) ```

## Arguments

 `Delta` vector with `Dmax`+1 components : weights in the penalty function. `n` integer : number of observatons. `p` integer : number of variables. `K` scalar : constant in the penalty function.

## Value

A vector with the same length as Delta: for each `d`=0, ..., `Dmax`, let `N`=`n`-`d`, `D`=`d+1` and
`pen(d) = x K N/(N-1)` where `x` satisfies

φ`(x) = exp(-Delta(d))`, when `Delta(d)<50`,
where φ`(x)=pf(q=x/(D+2),df1=D+2,df2=N-1,lower.tail=F)-(x/D)pf(q=(N+1)x/D(N-1),df1=D,df2=N+1,lower.tail=F)`

ψ`(x) = Delta(d)`, when `Delta(d)``50`,
where ψ```(x)=lbeta(1+D/2,(N-1)/2)-log(2(2x+(N-1)D)/((N-1)(N+2)x))-(N-1)/2log((N-1)/(N-1+x))-(D/2)log(x/(N-1+x)) ```

## Note

The values of the penalty function greater than 1e+08 are set to 1e+08.

If for some `Delta(d)` the equation φ`(x) = exp(-Delta(d)/(d+1))` has no solution, then the execution is stopped.

## Author(s)

Yannick Baraud, Christophe Giraud, Sylvie Huet

LINselect documentation built on Jan. 10, 2020, 9:08 a.m.