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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
  • $\begingroup$ @JohnK Thus, in your opinion, it is enough to take those cases as given? Are there other cases in which we are sure a UMP test does not exist? $\endgroup$ – PhDing Jan 2 '16 at 8:43
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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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