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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.

user234
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  • Strictly speaking your $\lambda$ should be a product over the $n$ observations – Henry Jan 03 '18 at 21:58
  • Assuming all your $X_i$ are non-negative, your likelihood ratio can only take three values: a constant positive value when $\max X_i \lt \theta_0$, a zero value when $\theta_0 \le \max X_i \lt \theta_1$ and an undefined value when $\max X_i \ge \theta_1$. This rather undermines the Neyman-Pearson Lemma, since it only gives you a way forward for $\alpha=0$ or $1$. For other $\alpha \in (0,1)$ there will be several most powerful tests of size $\alpha$, not a unique one – Henry Jan 03 '18 at 22:02

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