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I think I understand the Neyman-Pearson lemma, but I'm really struggling to understand the reasoning with which it's used as a building block to build tests for composite hypotheses.

Take this worked example, say. At the end, they say that "the" critical region C defines a UMP test, but from what I can tell, they've got a whole family of regions C, one for each alternative hypothesis $\mu_\alpha$. So you still can't say you've found a single test which is UMP for the entrire alternative hypothesis $\mu_\alpha > 10.

Jack M
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  • There's just one critical region specified there: $C=\{(x_1,x_2,\ldots,x_n):\bar{x} \geq k^*\}$. The critical value $k^*$ is defined solely by considering the simple null hypothesis. – Scortchi - Reinstate Monica Dec 22 '16 at 17:25
  • @Scortchi Let me get this straight. So for each $\mu$, and each $k$, we can find the region $C(\mu, k)$ such that $L(10)/L(\mu) – Jack M Dec 23 '16 at 18:30
  • We got lucky - the Gaussian distribution has the monotone likelihood ratio property. – Scortchi - Reinstate Monica Dec 23 '16 at 20:06
  • @Scortchi So when that property isn't satisfied, is finding a UMP test just not possible? What if you apply Neyman-Pearson to find an optimal region for each alternative hypothesis $\theta\in\Theta_1$, and find that they're all different for each $\theta$? Is the conclusion that there doesn't exist a UMP test for the composite alternative hypothesis $\Theta_1$? – Jack M Dec 23 '16 at 20:14
  • That's right. Try that with e.g. tests on the location parameter of a Cauchy (with known scale) based on a single observation. – Scortchi - Reinstate Monica Dec 23 '16 at 20:20
  • For testing $H_0:\theta=\theta_0$ against $H_1:\theta \in \Theta_1\subset \Theta-\{\theta_0\}$, suppose one can derive an MP test using NP lemma by expressing the alternative as $H_1:\theta=\theta_1$ where $\theta_1$ is any member of $\Theta_1$. If this test is independent of $\theta_1$, then it is also UMP by definition; if it depends on $\theta_1$, then no UMP test exists. And as already mentioned, if the population distribution has the MLR property then a UMP test is found using Karlin-Rubin theorem. These two methods are discussed [here](https://stats.stackexchange.com/q/158698/119261). – StubbornAtom Jul 22 '20 at 18:13
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    Does this answer your question? [Ways to find a UMP test](https://stats.stackexchange.com/questions/158698/ways-to-find-a-ump-test) – StubbornAtom Jul 22 '20 at 18:17

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