
likelihood ratio test
Let x have the distribution with pdf $\displaystyle f = \frac {1}{2} \exp {x\theta} $
Let $\displaystyle X_1...X_5 $ be ordered statistic, x between inf and inf
Find the likelihood ratio for testing the hypothesis $\displaystyle H_0: \theta=\theta_0 $ against$\displaystyle H_a : \theta not = \theta_0 $
likelihood function= $\displaystyle \Pi \frac {1}{2} \exp {x\theta} $
Not really sure how to continue?
Any help will be appreciated
Thanks in advance

Not sure here.
The Variable
$\displaystyle D = 2\left( ln L(\theta_0)  ln L(\hat{\theta} \right) $
follows a chisquare distribution with $\displaystyle df_1  df_2 $ degrees of freedown of each model. The likelihoods are maximized under the null hypothesis ($\displaystyle \theta_0 $) and under the alternative (MLE).
Since you already the likelihood, you know $\displaystyle L(\theta_0) $, as $\displaystyle \theta_0 $ is given. Finally, $\displaystyle \hat{\theta} $ is just the MLE (the point where the likelihood is a maximum) and then the variable $\displaystyle D $ can be obtained.
Again, not 100% sure...

Should i split the likelihood ratio into 2 regions, 1 is $\displaystyle x > \theta $ and other one is $\displaystyle x < \theta $ ?
for $\displaystyle x > \theta $, MLE is $\displaystyle \theta = X_{(1)} $
and
for $\displaystyle x < \theta $, MLE is $\displaystyle \theta = X_{(5)} $?
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! :)