📂Mathematical Statistics

Power of a Nuisance Test and the Most Powerful Test

Definition ¹

Hypothesis Testing: $$ \begin{align*} H_{0} :& \theta \in \Theta_{0} \\ H_{1} :& \theta \in \Theta_{0}^{c} \end{align*} $$

1. A power function $\beta (\theta)$ is said to be unbiased if it satisfies the following for all $\theta_{0} \in \Theta_{0}$ and $\theta_{1} \in \Theta_{0}^{c}$: $$ \beta \left( \theta_{0} \right) \le \beta \left( \theta_{1} \right) $$
2. Let $\mathcal{C}$ be a set comprising such hypothesis tests. A hypothesis test $A$ that has a power function $\beta (\theta)$, among the hypothesis tests in $\mathcal{C}$, satisfies the following for all $\theta \in \Theta_{0}^{c}$ and for any power function $\beta ' (\theta)$ of all hypothesis tests in $\mathcal{C}$: $$ \beta ' (\theta) \le \beta (\theta) $$ is called a (Uniformly) Most Powerful Test, UMP.

Explanation

Unbiased Power Function

$$ \beta (\theta) := P_{\theta} \left( \mathbf{X} \in \mathbb{R} \right) $$ The power function varies depending on the probability $P$, precisely, the probability distribution of $X$ and the rejection region $R$; hence, it’s difficult to fully visualize the form of $\beta$ based on the definition alone. Nonetheless, a sensibly good power function should ideally have higher detection power under the alternative hypothesis than under the null hypothesis. This property of maintaining higher detection power irrespective of how $\theta_{0}$ and $\theta_{1}$ are chosen is referred to as the unbiasedness of the power function. This concept of comparing power function values leads to the concept of the most powerful test discussed next.

Most Powerful Test

Most powerful… Not something straight out of a thrilling boy’s comic, but literally the strongest hypothesis test.

From the definition’s statement, for a test to be the most powerful means that for all $\theta \in \Theta_{0}^{c}$ where the null hypothesis rightly should be rejected, no other power function $\beta '$ can surpass the detection power of the most powerful test’s power function $\beta$.