a short word on profile likelihood

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 confidenceinterval,

> 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)

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong> <embed style="" type="" id="" height="" width="" src="" object=""> <iframe width="" height="" frameborder="" scrolling="" marginheight="" marginwidth="" src=""> <object style="" height="" width="" param="" embed=""> <param name="" value="">