# likelihood ratio test

• Aug 5th 2010, 05:29 PM
firebio
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

• Aug 6th 2010, 01:25 AM
gustavodecastro
Not sure here.

The Variable

$\displaystyle D = -2\left( ln L(\theta_0) - ln L(\hat{\theta} \right)$

follows a chi-square 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...
• Aug 6th 2010, 07:12 AM
firebio
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
• Aug 6th 2010, 01:04 PM
gustavodecastro
I've changed my post because I misunderstood your question. Sorry for that.