I was given the following question in a practice exam. $X\sim f(x;\theta)=\theta e^{-\theta x}I(0<x<\infty)$ with the following hypothesis $H:\theta \ne 1 \text{ vs } A: \theta=1$ The question asks to find a UMP test of size $\alpha=.05$
I figured that since the distribution is exponential then it has a monotone likelihood ratio in $X$. If I consider the one tail test $H: \theta>1$ vs $A: \theta=1$ then by the Neymar pearson lemma there exists a UMP test of size $\alpha=.05$ of the form $X<t_1$ where $t_1$ is chosen to produce an $\alpha=.05$. Note since our original hypothesis is two tail, then the one tail test will beat ours in power;therefore, there cannot exist a UMP test for this hypothesis. The best we can hope for is an UMPU Test. Is this correct.