MAT8886 a short word on profile likelihood

February 7, 2012

(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

Profile likelihood is an interesting theory to visualize and compute confidence interval for estimators (see e.g. Venzon & Moolgavkar (1988)). As we will use is, we will plot

But more generally, it is possible to consider

where Then (under standard suitable conditions)

which can be used to derive confidence intervals.
> base1=read.table(
+ "",
+ header=TRUE)
> library(evir)
> X=base1$
> u=5

The function to draw the profile likelihood for the tail index parameter is then

> Y=X[X>u]-u
> loglikelihood=function(xi,beta){
+ sum(log(dgpd(Y,xi,mu=0,beta))) }
> XIV=(1:300)/100;L=rep(NA,300)
> for(i in 1:300){
+ XI=XIV[i]
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ L[i]=-optim(par=1,fn=profilelikelihood)$value }
> plot(XIV,L,type="l")

It is possible to use it that profile likelihood function to derive a confidence interval,

> PL=function(XI){
+ profilelikelihood=function(beta){
+ -loglikelihood(XI,beta) }
+ return(optim(par=1,fn=profilelikelihood)$value)}
> (OPT=optimize(f=PL,interval=c(0,3)))
[1] 0.6315989
[1] 754.1115
> up=OPT$objective
> abline(h=-up)
> abline(h=-up-qchisq(p=.95,df=1),col="red")
> I=which(L>=-up-qchisq(p=.95,df=1))
> lines(XIV[I],rep(-up-qchisq(p=.95,df=1),length(I)),
+ lwd=5,col="red")
> abline(v=range(XIV[I]),lty=2,col="red")

This is done with the following code

> library(ismev)
> gpd.profxi(,5),xlow=0,xup=3)

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