Consider some simulated data
> set.seed(1)
> x=exp(rnorm(100))
Assume that those data are observed i.id. random variables with distribution
, with )
. The natural idea is to consider the maximum likelihood estimator
\right\rbrace)
For instance, consider some maximum likelihood estimator,
> library(MASS)
> (F=fitdistr(x,"gamma"))
shape rate
1.4214497 0.8619969
(0.1822570) (0.1320717)
> F$estimate[1]+c(-1,1)*1.96*F$sd[1]
[1] 1.064226 1.778673
Here, we have an approximated (since the maximum likelihood has an asymptotic Gaussian distribution) confidence interval for 
. We can use numerical optimization routine to get the maximum of the log-likelihood function
> log_lik=function(theta){
+ a=theta[1]
+ b=theta[2]
+ logL=sum(log(dgamma(x,a,b)))
+ return(-logL)
+ }
> optim(c(1,1),log_lik)
$par
[1] 1.4214116 0.8620311
$value
[1] 146.5909
And we have the same value.
Now, what if we care only about , and not 
. The we can use profile likelihood. The idea is to solve
\right\rbrace%20\right\rbrace)
i.e.
\right\rbrace)
or, equivalently,
\right\rbrace)
> prof_log_lik=function(a){
+ b=(optim(1,function(z) -sum(log(dgamma(x,a,z)))))$par
+ return(-sum(log(dgamma(x,a,b))))
+ }
> vx=seq(.5,3,length=101)
> vl=-Vectorize(prof_log_lik)(vx)
> plot(vx,vl,type="l")
> optim(1,prof_log_lik)
$par
[1] 1.421094
$value
[1] 146.5909
A few weeks ago, we have mentioned the likelihood ratio test, i.e.
-\log\mathcal{L}(\theta_0)]\sim%20n[\widehat{\theta}-{\theta}_0]^2I(\theta_0)\sim%20\chi^2(k))
The analogous can be obtained here, since
-\log\mathcal{L}_p(\alpha_0)]\sim\chi^2(1))
(the 1 comes from the fact that is a one-dimensional coefficient). The (technical) proof can be found in Suhasini Subba Rao’s notes (see also Section 4.5.2 in Antony Davison’s Statistical Models). From that property, we can easily obtain a confidence interval for
Hence, from our sample, we get the following 95% confidence interval,
> abline(v=optim(1,prof_log_lik)$par,lty=2)
> abline(h=-optim(1,prof_log_lik)$value)
> abline(h=-optim(1,prof_log_lik)$value-qchisq(.95,1)/2)
> segments(F$estimate[1]-1.96*F$sd[1],
-170,F$estimate[1]+1.96*F$sd[1],-170,lwd=3,col="blue")
> borne=-optim(1,prof_log_lik)$value-qchisq(.95,1)/2
> (b1=uniroot(function(z) Vectorize(prof_log_lik)(z)+borne,c(.5,1.5))$root)
[1] 1.095726
> (b2=uniroot(function(z) Vectorize(prof_log_lik)(z)+borne,c(1.25,2.5))$root)
[1] 1.811809
that can be visualized below,
> segments(b1,-168,b2,-168,lwd=3,col="red")
In blue the obtained obtained using the asymptotic Gaussian property of the maximum likelihood estimator, and in red, the obtained obtained using the asymptotic chi-square distribution of the log (profile) likelihood ratio.
