📂Mathematical Statistics

Hypothesis Testing and the One-to-One Correspondence of Confidence Sets

Theorem

Let’s assume we have parameter space $\Theta$ and space $\mathcal{X}$ given.

• For each $\theta_{0} \in \Theta$, let $A \left( \theta_{0} \right)$ be the rejection region $\alpha$ of the hypothesis test $H_{0} : \theta = \theta_{0}$. For each $\mathbf{x} \in \mathcal{X}$, let’s define the set $C \left( \mathbf{x} \right) \subset \Theta$ as follows: $$ C \left( \mathbf{x} \right) := \left\{ \theta_{0} : \mathbf{x} \in A \left( \theta_{0} \right) \right\} $$ Then, the random set $C \left( \mathbf{X} \right)$ is a $1 - \alpha$ confidence set.
• Conversely, if $C \left( \mathbf{X} \right)$ is a $1 - \alpha$ confidence set, for all $\theta_{0} \in \Theta$ let’s define the set $A \left( \theta_{0} \right) \subset \mathcal{X}$ as follows: $$ A \left( \theta_{0} \right) = \left\{ \mathbf{x} : \theta_{0} \in C \left( \mathbf{x} \right) \right\} $$ Then, the event $A \left( \theta_{0} \right)$ is the rejection region $\alpha$ of the hypothesis test $H_{0} : \theta = \theta_{0}$.

Explanation

To briefly summarize the motivation behind this theorem, it is as follows: $$ \theta_{0} \in C \left( \mathbf{x} \right) \iff \mathbf{x} \in A \left( \theta_{0} \right) $$

Proof ¹

$\left( \implies \right)$

Since $A \left( \theta_{0} \right)$ is the rejection region of level $\alpha$, $$ \begin{align*} P_{\theta_{0}} \left( \mathbf{X} \notin A \left( \theta_{0} \right) \right) \le & \alpha \\ P_{\theta_{0}} \left( \mathbf{X} \in A \left( \theta_{0} \right) \right) \ge & 1 - \alpha \end{align*} $$ As it holds for all $\theta_{0}$ given the assumption, we can write it as $\theta$, and since we defined $C \left( \mathbf{x} \right) = \left\{ \theta_{0} : \mathbf{x} \in A \left( \theta_{0} \right) \right\}$, the coverage probability of $C \left( \mathbf{X} \right)$ is $$ P_{\theta} \left( \mathbf{X} \in C \left( \mathbf{X} \right) \right) = P_{\theta} \left( \mathbf{X} \in A \left( \theta \right) \right) \ge 1 - \alpha $$ In other words, $C \left( \mathbf{X} \right)$ is a $1-\alpha$ confidence set.

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$\left( \impliedby \right)$

The probability of a type I error for $A \left( \theta_{0} \right)$ in $H_{0} : \theta = \theta_{0}$ is $$ P_{\theta_{0}} \left( \mathbf{X} \notin A \left( \theta_{0} \right) \right) = P_{\theta_{0}} \left( \theta_{0} \notin C \left( \mathbf{X} \right) \right) \le \alpha $$ Therefore, it is a level $\alpha$ hypothesis test.

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