Hypothesis

Definition

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

Suppose due to theory/preliminary experiment

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

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