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I have a sample $X=(X_1, ...,X_n)\sim N(\mu,\sigma^2)$ with $\sigma^2$ known. The hypotheses are $H_0: \mu=\mu_0, H_1:\mu \neq \mu_0$. I know that in such a case an UMP test does not exist and so that I should proceed using a LR test, in order to find the rejection rule.

My professor also told me that for a sample distribution that belongs to the Exponential Family in the case of simple vs bilateral hypotheses an UMP test does not exist.

Thus, my question is theoretic: why does an UMP test does not exist in such cases? Which are the conditions under which an UMP test does not exist?

EDIT: I have found an example in which, instead, although the alternative hypothesis is bilateral, the UMP test exists.

A sample $X\sim U(0,\theta)$. The hypotheses are $H_0:\theta=\theta_0, H_1:\theta\neq \theta_0$.

Lehman, 1986 p.111

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    $\begingroup$ Simply put, a UMP test does not exist because the critical regions turn out to be different for $\theta> \theta_0$ and $\theta<\theta_0$. This means that there are only UMP tests for one-sided hypotheses, where you may still use the NP lemma. $\endgroup$ – JohnK Dec 30 '15 at 21:54
  • $\begingroup$ @JohnK Ok, that's clear. But how can I apply this idea to the two examples above? How can I show that it is actually the case? $\endgroup$ – PhDing Dec 31 '15 at 8:57
  • $\begingroup$ I am not familiar with a general proof. Perhaps you can work something out starting from the definition of an exponential density. My guess is that it would be non-trivial. $\endgroup$ – JohnK Dec 31 '15 at 10:57
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    $\begingroup$ @Alessandro: it depends on how deep you want to dive into it. I think the paper is interesting to learn which conditions are needed and why $\endgroup$ – user83346 Jan 2 '16 at 12:10
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    $\begingroup$ I thought you were interested in exponential densities. The uniform distribution does not belong to the exponential family so there is no general rule there. $\endgroup$ – JohnK Jan 3 '16 at 16:51
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In the example that you have provided, go on to calculate the likelihood ratio and you will find that it comes out to be a function of the order statistics, X(1) and X(2). Question 8.33 of Statistical Inference by George and Casella will help. The solution is provided in the link below: http://www.ams.sunysb.edu/~zhu/ams570/Solutions-Casella-Berger.pdf

Coming back to the existence of a UMP test, Karlin Rubin Theorem tells that the MLR should exist, so that the inverse operation can be applied to get the test. The example on the link below will surely help. http://web.eecs.umich.edu/~cscott/past_courses/eecs564w11/25_ump.pdf

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