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 $H_0:\theta=\theta_0, H_1:\theta\neq \theta_0$.
Best Answer
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