Given a sample of size n from $Unif(0,\theta)$, does there exist a most powerful test of the hypotheses: $H_0: \theta = \theta_0$ vs $H_1: \theta = \theta_1$, where $ \theta_0 > \theta_1?$
I tried applying the Neyman-Pearson Lemma to get:
$ \lambda = \frac{\frac{1}{\theta_0^n}.1._{[X_n:n < \theta_0}]}{\frac{1}{\theta_1^n}.1._{[X_n:n < \theta_1}]} $
Then would an MP test be to reject if $\lambda < c$ for some $c$, even though $\lambda$ is independent of $X_i$?
Any clarification would be greatly appreciated.