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I am confused about the last highlighted sentence regarding finding a subset S, for which BOTH Type I and Type II error probabilities are 0. For $P_\theta S$, which is the probability with which we reject the null hypothesis $H_0$, if the actual $\theta$ says that our $H_0$ is true, then shouldn't the probability of rejecting $H_0$ be 0 and not 1? Should I be imagining $S$ to be $C^c$ from the definition of a critical region?

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You are right, it looks like 0 and 1 have got accidentally changed in the last highlighted sentence. If $P_\theta S$ is the probability of the critical region, it should ideally be 0, when $\theta\in \Theta_0$ ($H_0$ true) and it should ideally be 1, $\theta\in \Theta_1$ ($H_0$ false). Note that the probability of type II error was $P_\theta S^C=1-P_\theta S$, $\theta \in \Theta_1$ ($S^C$ is the complement of $S$). So, now both errors would be 0.

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