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Math Help - likelihood ratio test

  1. #1
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    likelihood ratio test

    Let x have the distribution with pdf  f = \frac {1}{2} \exp {-|x-\theta|}
    Let  X_1...X_5 be ordered statistic, x between -inf and inf

    Find the likelihood ratio for testing the hypothesis  H_0: \theta=\theta_0 against  H_a : \theta  not = \theta_0

    likelihood function=  \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

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

    follows a chi-square distribution with  df_1 - df_2 degrees of freedown of each model. The likelihoods are maximized under the null hypothesis (  \theta_0 ) and under the alternative (MLE).

    Since you already the likelihood, you know  L(\theta_0) , as  \theta_0 is given. Finally,  \hat{\theta} is just the MLE (the point where the likelihood is a maximum) and then the variable  D can be obtained.

    Again, not 100% sure...
    Last edited by gustavodecastro; August 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  x > \theta and other one is  x <  \theta ?

    for  x > \theta , MLE is  \theta = X_{(1)}
    and
    for  x < \theta , MLE is  \theta = X_{(5)} ?


    Thanks
    Last edited by firebio; August 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; August 6th 2010 at 09:50 PM.
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