Hypothesis

Definition

Let $X$ : Random variable with pdf : $f(x;\theta ),\theta \in \Omega$

Suppose due to theory/preliminary experiment

• $\theta \in {\omega }_0$ or $\theta \in {\omega }_1$
• $w_0,w_1$ disjoint subsets of $\Omega$
• ${\omega }_0\cup {\omega }_1=\Omega$

Then

• We label these hypotheses as $$\begin{array}{rl}{H}_0& :\theta \in {\omega }_0\\ H_1& :\theta \in {\omega }_1\end{array}$$
• We refer the hypothesis $H_0$ as the null hypothesis
• We refer the hypothesis $H_1$ as the alternative hypothesis