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I need to obtain the functional (i.e.: statistic obtained from the empirical distribution function) associated to the most powerful test to contrast $H_0: f_o$ vs. $H_1: f_1$, using a sample of size $n$.

In particular, my problem is that I don’t know how to construct this functional when I am not given a particular density function.

Using the Neyman Pearson lemma, I know that the maximum power test can be performed by computing the likelihood ratio test as:

$$ \Lambda(x_1,...,x_n)=\dfrac{{\sup}f_0(x_1,...,x_n)} {{\sup}f_1(x_1,...,x_n)} $$

However since I don’t have a specific density function, I don’t know if this expression can be further operated or simplified in any way.

EDIT: Since I got no answers or comments, my guess is that there may be something wrong with my question.

EDIT 2: Per comments I understand now that I should use non parametric statistics and instead of conventional maximum likelihood, the appropriate ratio to use should be based on empirical likelihood.

In the introduction of this paper: Empirical Likelihood Based Hypothesis Testing, there is a derivation of an statistic based on Empirical likelihood ratio to test $H_0: F = F_0$. However, since my data do not have the prerequisite of $F$ being a continuous distribution I am not sure if such reasoning can be applied to my original question.

kjetil b halvorsen
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sets
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  • It is not clear what your question is! Did you read about empirical likelihood? – kjetil b halvorsen Jul 24 '13 at 11:34
  • @kjetilbhalvorsen, thanks for your comment. My question is simpy how to derive the mathematical expression for the maximum power test $H_0: f_o$ vs. $H_1: f_1$ based on a sample of size $n$. The rest of the "information" above is just my take on the possible answer, which I guess is wrong at some point. – sets Jul 24 '13 at 11:42
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    You write: "...when I am not given a particular density function. ...". Is this homework, since you say "not given"? To compute a likelihood ratio test, you need a model. What is your model? If your model is "non-parametric", you could read http://en.wikipedia.org/wiki/Empirical_likelihood – kjetil b halvorsen Jul 24 '13 at 15:10
  • @kjetilbhalvorsen. thanks again! this is indeed self study as tagged in the question. i have read your link and some other articles regarding empirical likelihood, but still I am not sure how to proceeed. I will update the original question – sets Jul 26 '13 at 07:03

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