MAT8886 a short word on profile likelihood

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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

https://i2.wp.com/freakonometrics.blog.free.fr/public/perso5/proflike01.gif?w=578

But more generally, it is possible to consider

https://i1.wp.com/freakonometrics.blog.free.fr/public/perso5/profilik06.gif?w=578


where https://i2.wp.com/freakonometrics.blog.free.fr/public/perso5/profilik03.gif?w=578. Then (under standard suitable conditions)

https://i1.wp.com/freakonometrics.blog.free.fr/public/perso5/profilik05.gif?w=578


which can be used to derive confidence intervals.

> base1=read.table(
+ "http://freakonometrics.free.fr/danish-univariate.txt",
+ header=TRUE)
> library(evir)
> X=base1$Loss.in.DKM
> 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)))
$minimum
[1] 0.6315989
 
$objective
[1] 754.1115
> up=OPT$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(gpd.fit(X,5),xlow=0,xup=3)

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