# What is the general methodology for constructing a UMP test for a simple hypothesis versus a composite one?

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 $\mu_\alpha > 10. • There's just one critical region specified there:$C=\{(x_1,x_2,\ldots,x_n):\bar{x} \geq k^*\}$. The critical value$k^*$is defined solely by considering the simple null hypothesis. Commented Dec 22, 2016 at 17:25 • @Scortchi Let me get this straight. So for each$\mu$, and each$k$, we can find the region$C(\mu, k)$such that$L(10)/L(\mu)<k$. That region depends on both$\mu$and$k$. Now we ask: which value of$k$will give that region a fixed probability$\alpha$? That value of$k$will be different for each$\mu$, call it$k(\mu)$. And incredibly,$C(\mu, k(\mu))$is constant as a function of$\mu$, even though$k$might not be! Does that witchcraft always happen, or did we just get lucky because of algebraic properties of the Gaussian distribution? Commented Dec 23, 2016 at 18:30 • We got lucky - the Gaussian distribution has the monotone likelihood ratio property. Commented Dec 23, 2016 at 20:06 • @Scortchi So when that property isn't satisfied, is finding a UMP test just not possible? What if you apply Neyman-Pearson to find an optimal region for each alternative hypothesis$\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\$? Commented Dec 23, 2016 at 20:14
• Does this answer your question? Ways to find a UMP test Commented Jul 22, 2020 at 18:17