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I've just read answer to this question about finding UMP tests. Part of the question concerns possibility of using the same test as from Neymann-Pearson lemma also for composite hypotheses by replacing $H_0: \theta = \theta_0$ vs $H_1: \theta > \theta_0$ with $H_0: \theta = \theta_0$ vs $H_1: \theta = \theta_1$ with assumption that $\theta_1 > \theta_0$.

(1) Part & parcel of being a uniformly most powerful test for $H_0:θ=θ_0$ vs $H_1:θ>θ_0$ is being most powerful for $H_0:θ=θ_0$ vs $H1:θ=θ_1$ for whichever $θ_1>θ$ you choose. So the tests are exactly the same. (But there isn't always a UMP test for one-sided alternative hypotheses. Testing hypotheses about the location parameter of a Cauchy with known scale is a standard example.)

and I'm not sure about some details:

  1. Isn't it true only in cases when from calculations it turns out that formula for critical region (or maybe formula after expanding $\alpha = P(X \in C | H_0)$, I'm not sure) doesn't contain $H_1$ term?

  2. Can it be used for two-sided hypothesis too? i.e. $H_0: \theta \leq \theta_0$ vs $H_1: \theta > \theta_0$

Brzoskwinia
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  • The term "two-sided" is not normally used as you use it, but rather for tests of $H_0:\ \theta=\theta_0$ vs. $H_1:\ \theta\neq \theta_0$, i.e., where the alternative is on both sides of the null hypothesis. – Christian Hennig Nov 13 '21 at 23:24

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