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Thread: likelihood ratio test

  1. #1
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    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
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  2. #2
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    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...
    Last edited by gustavodecastro; Aug 6th 2010 at 12:33 PM. Reason: There was a mistake in the first equation
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  3. #3
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    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
    Last edited by firebio; Aug 6th 2010 at 07:36 AM.
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  4. #4
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    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!
    Last edited by gustavodecastro; Aug 6th 2010 at 09:50 PM.
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