I have to find the uniformly most powerful size $\alpha$ test for the following pdf- $$f(x|\theta) = exp\{-(x-\theta)\}$$ $x>\theta$
The pdf is 0 for $x\leq\theta$ , Null hypothesis is $\theta = \theta_0$ and alternate is $\theta =\theta_1$. I am given an i.i.d. sample from the above pdf. I was trying to apply the Neyman pearson lemma but the relative likelihood is constant so I don’t know how to find the UMP size $\alpha$ test, can someone please provide the solution.