penalty {LINselect}

penalty

Description

Calculate the penalty function for estimators selection.

Usage

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

\phi(x) = exp(-Delta(d)), when Delta(d)<50,
where \phi(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)

\psi(x) = Delta(d), when Delta(d)\ge50,
where \psi(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 \phi(x) = exp(-Delta(d)/(d+1)) has no solution, then the execution is stopped.

Author(s)

Yannick Baraud, Christophe Giraud, Sylvie Huet