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}

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