
likelihood ratio test
Let x have the distribution with pdf
Let be ordered statistic, x between inf and inf
Find the likelihood ratio for testing the hypothesis against
likelihood function=
Not really sure how to continue?
Any help will be appreciated
Thanks in advance

Not sure here.
The Variable
follows a chisquare distribution with degrees of freedown of each model. The likelihoods are maximized under the null hypothesis ( ) and under the alternative (MLE).
Since you already the likelihood, you know , as is given. Finally, is just the MLE (the point where the likelihood is a maximum) and then the variable can be obtained.
Again, not 100% sure...

Should i split the likelihood ratio into 2 regions, 1 is and other one is ?
for , MLE is
and
for , MLE is ?
Thanks

I've changed my post because I misunderstood your question. Sorry for that.
About your post, I think your are right, but perhaps someone can be certain! :)