3.3 p-valuees

• $\alpha \in [0, 1]$,

\[S_{\alpha} \subseteq S_{\alpha^{\prime}} \quad \alpha < \alpha^{\prime}\] \[\hat{p}(X) := \inf \{ \alpha \mid X \in S_{\alpha} \}\] \[\phi_{\alpha} := \{ x \in \mathcal{X} \mid \}\]

Definition p-values

\[S_{\alpha} := \{ x \in \mathcal{X} \mid \phi_{\alpha}(x) = 1 \}\] \[\hat{p}(x) := \inf \{ \alpha \mid x \in S_{\alpha} \}\]

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Example 3.3.1

Lemma 3.3.1

• $X$,
• \(P_{\theta}\),
• $\Theta$,
• \(\Theta_{H}\),

(1) If

\[\sup_{\theta \in \Theta_{H}} P_{\theta}(X \in S_{\alpha}) \le \alpha \quad 0 < \forall \alpha < 1,\]

then $