Questions about significance level and Type II error

Question

I am learning tests of statistical hypotheses by myself. The book I used is Probability and Statistical Inference by Robert V. Hogg, etc.

I want to confirm some of my understandings are right or not.

$S=C\cup C',$ and $C\cap C'=\varnothing$

H0: null hypothesis

H1: alternative hypothesis

$C$: Critical region

1. I think that $\alpha$ and $\beta$ are inversely related. Here is my proof. If $\alpha$=P(type I error) increase, then it means $C$ get bigger than before. Therefore the $\beta$=P(type II error) decrease, since when $C$ get bigger, then $C'$ will get smaller.

   Moreover, if $\beta$=P(type II error) increase, then it means $C'$ get bigger than before. Therefore the $\alpha$=P(type I error) decrease, since when $C'$ get bigger, then $C$ will get smaller.

2. Just to confirm, when the result of our experiments fall into $C'$, we reject H1 (i.e. we accept H0), right?

Am I right? If I am wrong please correct me and tell me why I am wrong.

Answer 1

Suppose you collect i.i.d samples $X_1,\dots,X_n$ from an unknown distribution. Let's assume that the distribution comes from a parametric model $\{\mathbb P_{\theta}\}_{\theta \in \Theta \subset \mathbb R^k}$. We want to do inference on $\theta$. Make a partition $\Theta = \Theta_0 \sqcup \Theta_1$. Our goal is to decide if $\theta \in \Theta_0$, which is known as the null hypothesis $H_0$, or if $\theta \in \Theta_1$, which is known as the alternative hypothesis $H_1$. The idea is that we'll make some rule on when to reject $H_0$. The way this is done is by constructing a function of the data $T(X_1,\dots,X_N)$ called a test statistic, and we reject $H_0$ when $T\in R$ for some rejection region.

What are some desirable properties of $T$ and $R$? Obviously we want to minimize false positives (Type I error) and false negatives (Type II error). In symbols, we want to minimize $$\alpha \overset{\text{def}}{=}\sup_{\theta \in \Theta_0}\mathbb P_{\theta}(T\in R)$$ and to maximize $$\beta \overset{\text{def}}{=}\inf_{\theta \in \Theta_1}\mathbb P_{\theta}(T\in R)$$

The issue is that we can see minimizing $\alpha$ corresponds to shrinking $R$. But if $R$ shrinks, then $\beta$ also shrinks. Similarly, maximizing $\beta$ corresponds to expanding $R$, but then $\alpha$ increases.