Test

Definition

Let

• $X$ : Random variable wth pdf $f(x;\theta ),\theta \in \Omega$
• $X_1,\dots ,X_n$ : Random sample of $X$
• $\mathcal{D}=\mathrm{space}\{(X_1,\dots ,X_n)\}$ : Sample space

Suppose

• $H_0$ : Null hypothesis
• $H_1$ : Alternative hypothesis

Then

• A test of $H_0$ versus $H_1$ is based on a subset $C\subset \mathcal{D}$
• We call $C$ the critical region
• Decision rule (test) of $C$ is

$$\begin{array}{rl}\text{Reject }{H}_0(\text{Accept }{H}_1)& \text{if }(X_1,\dots ,X_n)\in C\\ \text{Retain }{H}_0(\text{Reject }{H}_1)& \text{if }(X_1,\dots ,X_n)\in C^c\end{array}$$