Uniformly most powerful test for two-sided hypothesis

Question

I need some help:

Prove that a uniformly most powerful test for a level $\alpha\in(0,1)$ doesn't exist for the test $H_0:\mu=\mu_0$ versus $H_1:\mu\neq\mu_0$, while $\mu,\mu_0\in\mathbb{R}$.

Answer 1

It's not true in general. so you can not prove it. according to

Uniformly Most Powerful Test for a Uniform Sample

UMP test for

$\left\{\begin{array}{cc} H_0: & \theta= \theta_0 \\ H_1: & \theta \neq \theta_0 \end{array} \right.$

exist. ( $\{x_i \}_{i=1}^{n} \overset{i.i.d}{\sim} Uniform(0,\theta)$)