📂Mathematical Statistics

Most Accurate Confidence Set

Definition ¹

For the hypothesis test for $\theta$, the confidence set of $1 - \alpha$ is called $C \left( \mathbf{x} \right)$, and let the acceptance region be $A \left( \theta \right) = C \left( \mathbf{x} \right)^{c}$.

1. The probability of False Coverage for $P_{\theta} \left( \theta’ \in C \left( \mathbf{X} \right) \right)$ against $\theta’ \ne \theta$ is called. The original coverage probability of $P_{\theta} \left( \theta \in C \left( \mathbf{X} \right) \right)$ is referred to as the contrasting term, True Coverage probability.
2. If for all $\theta’ \ne \theta$, the confidence set $C \left( \mathbf{x} \right)$ has a false coverage probability that is less than or equal to $1-\alpha$, it is said to be Unbiased. In other words, it satisfies the following. $$ P_{\theta} \left( \theta’ \in C \left( \mathbf{X} \right) \right) \le 1 - \alpha $$
3. Among the confidence sets of $1-\alpha$, the one that minimizes the false coverage probability is called the Uniformly Most Accurate (UMA) Confidence Set, and when it is a confidence interval, it is also called Neyman-shortest.

Explanation

Confidence Interval

Especially when $C \left( \mathbf{X} \right)$ is a confidence interval, the false coverage probability is separately defined into three types as follows. $$ \begin{align*} C \left( \mathbf{X} \right) = \left[ L , U \right] \implies & P_{\theta} \left( \theta’ \in C \left( \mathbf{X} \right) \right) \qquad , \theta’ \ne \theta \\ C \left( \mathbf{X} \right) = \left[ L , \infty \right) \implies & P_{\theta} \left( \theta’ \in C \left( \mathbf{X} \right) \right) \qquad , \theta’ < \theta \\ C \left( \mathbf{X} \right) = \left( -\infty , U \right] \implies & P_{\theta} \left( \theta’ \in C \left( \mathbf{X} \right) \right) \qquad , \theta’ > \theta \end{align*} $$

Relationship with UMP Hypothesis Testing

According to the one-to-one correspondence between hypothesis testing and confidence sets, there should be a corresponding confidence set for the most powerful test, which is reasonably considered to be the Uniformly Most Accurate confidence set.

In the definition, minimizing the false coverage probability is also connected to the question of whether the hypothesis test is optimal. The corresponding optimality of confidence intervals means that their length is the shortest, which is why we desire the confidence interval to be the shortest at the same confidence level, hence it is convincible to call it Neyman-shortest.

Considering Neyman-Pearson Lemma or Karl Pearson Theorem, the robustness they guarantee was too limited. Most powerful tests were essentially one-sided tests, but the significance of UMA especially emphasizes in the case of two-sided tests.