5.2 Set Estimation

• $g: \Theta \rightarrow \mathbb{R}$,
• $\Theta$,

5.2.1 Confidence Sets

Definition 5.47 confidense set

• $g: \Omega \rightarrow G$
• $\eta := 2^{G}$,
• $R: \mathcal{X} \rightarrow \eta$,
• $\gamma \in [0, 1]$,
  • given

The function $R$ is said to be a coefficient $\gamma$ confidence set for $g(\Theta)$ if for every $\theta \in \Omega$,

• \(\{x \mid g(\theta) \in R(x)\}\) is measurale, and

\[P_{\theta}^{\prime}(g(\theta) \in R(X)) \ge \gamma .\]

The confidence set $R$ is said to be exact if

\[\forall \theta \in \Omega, \ P_{\theta}^{\prime}(g(\theta) \in R(X)) = \gamma .\]

The confiddece set $R$ is said to be conservative if

\[\forall \theta \in \Omega, \ \inf_{\theta \in \Omega} P_{\theta}^{\prime}(g(\theta) \in R(X)) > \gamma .\]

■

Proposition 5.48

• $g: \Omega \rightarrow G$

proof

$\Box$