## Hypothesis Testing Basics

$H_{0}$: null hypothesis $\theta\in\Theta_{0}$ v.s. $H_{1}$: alternative hypothesis $\theta\in\Theta_{1}$

Two kinds of errors

1. reject the truth: $\alpha=P(X\in W)$, $\theta\in\Theta_{0}$;
size/significance level $\alpha$, the smaller($\to0$), the better

2. accept the false: $\beta=P(X\notin W)=1-P(X\in W)$, $\theta\in\Theta_{1}$;
power $1-\beta=P(X\in W)$, $\theta\in\Theta_{1}$, the larger($\to1$), the better

The power function $g(\theta)=P_{\theta\in\Theta=\Theta_{0}+\Theta_{1}}(X\in W)$

P-value $P=P_{0}(T>C)$

$T$: the test statistic

$W=\{T>T_{0}\}$: region of rejection / critical region

$C:$ test statistic value calculated from data

$P<\alpha$: small probability event happens, reject $H_{0}$
(reason: under $H_0$, for any value $C \sim T$, $C$ should be small, thus the area of $T>C$, or $P=P_{0}(T>C)$ should be large.)

$P\geq\alpha$: do not reject $H_{0}$