Coverage from test inversion

Let $\mathsf{E}=((\mathcal{Y},\mathcal{G}),(Q_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be an experiment with nonempty $\mathrm{\Theta }$, and let $0\le \alpha \le 1$. For each $\eta \in \mathrm{\Theta }$, let $A_{\eta }\in \mathcal{G}$ be the acceptance event of a test of the point null $\theta =\eta$, on the full observation space, including any auxiliary randomization. Suppose $Q_{\eta }(A_{\eta })\ge 1-\alpha$ for every $\eta \in \mathrm{\Theta }$. Then $C(y):=\{\eta \in \mathrm{\Theta }:y\in A_{\eta }\}$ is a confidence set with coverage at least $1-\alpha$.

The probability bound is the point-null level condition written in terms of acceptance events. For a randomized test, a rejection-probability function on the original data space is not itself an acceptance event; the auxiliary randomization must be part of the observation before applying this statement.

For a deterministic test of a simple null $\{\eta \}$ with a nonempty alternative, the acceptance event $A_{\eta }$ is the complement of its rejection event, so its test function is ${\phi }_{\eta }={\mathbf{1}}_{A_{\eta }^c}$. By Test size and power, its size is $E_{\eta }{\phi }_{\eta }=Q_{\eta }(A_{\eta }^c)$; level $\alpha$ therefore gives $Q_{\eta }(A_{\eta })=1-E_{\eta }{\phi }_{\eta }\ge 1-\alpha$. For a randomized test, its rejection-probability function is not itself an event. Applying this theorem then requires the realized auxiliary randomizer to be part of the observation and the stated eventwise probability bound to hold on that full observation space.