[R] Solving for parameter values at likelihood points

[Rubén Roa-Ureta]

ComRades,
On Oct 16, 2006, I posted a question regarding how to find the parameter 
values that made the likelihood function take a value equal to 1/8th of 
the maximum likelihood value. There was no reply but I found a solution 
and would like to know if there is a better solution using the function 
uniroot(). I think with uniroot() I can get better parameter values at 
these bounds.
This is my solution, using the function nearest() from package GenKern.
The example concerns the marginal likelihood for the normal variance 
parameter when the mean is a nuisance parameter (see section 7.3 in 
Royall, 1997, Statistical evidence. A likelihood paradigm, Chapman & Hall).
Rubén

y<-c(-2.250357,10.723614,7.238959,9.179939,5.831603,9.301580,4.019388,9.777280,5.587761,3.600276,9.753381,5.154928,2.776350,9.940350,14.185712,4.281198,10.994180,9.333027,13.082535,2.067826,10.772551,-3.811529,5.446021,-5.333287,1.421544)
n <- length(y)
s.sq <- (1/(n-1))*sum((y-mean(y))^2)
sigma.sq <- seq(1,100,0.1)
Lik.M <- sigma.sq^(-(n-1)/2)*exp(-(n-1)*s.sq/(2*sigma.sq))
Rel.Lik.M <- Lik.M/max(Lik.M)
plot(sigma.sq,Rel.Lik.M,type="l")
abline(h=1/8)
arrows(y0=0.4,y1=0.18,x0=0,x1=12)
arrows(y0=0.4,y1=0.18,x0=62,x1=50)
library(GenKern)
cbind(sigma.sq,Rel.Lik.M)[Rel.Lik.M==1] #maximum likelihood estimate
max.idx <- nearest(Rel.Lik.M,1,outside=TRUE) # vector index value at the 
maximum likelihood
#finding the parameter value at 1/8th the max. likelihood to the left of 
the maximum
cbind(sigma.sq[1:max.idx],Rel.Lik.M[1:max.idx])[nearest(Rel.Lik.M[1:max.idx],0.125),]
#finding the parameter value at 1/8th the max. likelihood to the right 
of the maximum
cbind(sigma.sq[max.idx:length(sigma.sq)],Rel.Lik.M[max.idx:length(sigma.sq)])[nearest(Rel.Lik.M[max.idx:length(sigma.sq)],0.125),]