Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$.
Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: \theta_1 <0$ and $H_1: \theta_1 > 0$.
I think the test statistic should be $X_{(1)}$, the first order statistic, and it's distribution can be get from formula. But I wonder how to show that it's indeed UMPUT?