For a shifted exponential the minimum carries all the information about the location

Let $X_1,X_2,\dots,X_n$ be a random sample observed from exponential distribution with location parameter $\theta$. For testing $H_0:\theta=\theta_0$ against $H_1:\theta>\theta_0$, the UMP test rejects $H_0$, if:

where C is to be chosen suitably.