# MAT8886 a short word on profile likelihood

February 7, 2012
By

(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(
+ "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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