Statistical Test

Definition (Ost20 pp.116, Statistical Test, Null and Alternative Hypotheses). Let $p_{\theta }(y)$ be a Parametric Probabilistic Model. The space parameter $\Theta$ is partitioned into two disjoint subsets ${\Theta }_0$ and ${\Theta }_1$. A test hypothesis is a statement about the parameter governing $p_\theta(y)$inrelationtotheseparameterspacesubsets.Specifically,thestatement$\theta\in\Theta_0\Leftrightarrow H=0$isreferredtoasth{e}_{n}ullhypothesi{s}_{a}ndthestatement$\theta\in\Theta_!\Leftrightarrow H=1$$is referred to as the alternative hypothesis.

Remark. A statistical hypothesis is a statement about the parameter of a probabilistic model. In the following, we will use the subscript notations $p_{{\theta }_0}$ and $p_{{\theta }_1}$ to indicate that the parameter $\theta$ of the probabilistic model $p_{\theta }$ is an element of ${\Theta }_0$ or ${\Theta }_1$, respectively.

Remark. The term null hypothesis is not necessarily the statement that some parameter assumes the value zero, even if this is often the case in practice. Rather, the null hypothesis in a statistical testing problem is the statement about the parameter one is willing to nullify, i.e., reject. Finally, the expressions $H=0$ and $H=1$ are not conceived as realizations of a random variable and hence hypothesis-Conditional Probability statements are not meaningful. The statements $H=0$ and $H=1$ are merely equivalent expressions for $\theta \in {\Theta }_0$ and $\theta \in {\Theta }_1$, respectively: $H=0$ refers to the true, but unknown, state of the world that the null hypothesis is true and the alternative hypothesis is false ($\theta \in {\Theta }_0$), and $H=1$ refers to the true, but unknown, state of the world that the alternative hypothesis is true and the null hypothesis is false ($\theta \in {\Theta }_1$).