Confidence Sets and Inversion

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment as in Statistical experiments, and assume $\mathrm{\Theta }\mathrm{\ne }\mathrm{\varnothing }$. A confidence-set procedure is a map $C:\mathcal{X}\to \mathcal{P}(\mathrm{\Theta })$, where $\mathcal{P}(\mathrm{\Theta }):=\{A:A\subseteq \mathrm{\Theta }\}$, such that, for every $\eta \in \mathrm{\Theta }$, its membership section $C_{\eta }:=\{x\in \mathcal{X}:\eta \in C(x)\}$ belongs to $\mathcal{A}$. Its coverage at $\theta$ is $c_C(\theta ):=P_{\theta }(C_{\theta })$, and its confidence level is ${\inf }_{\theta \in \mathrm{\Theta }}{c}_C(\theta )$, the greatest lower bound of those coverage values. For $0\le \alpha \le 1$, it has coverage at least $1-\alpha$ if $P_{\theta }(C_{\theta })\ge 1-\alpha$ for every $\theta \in \mathrm{\Theta }$.

Only membership sections need to be measurable; no sigma-algebra on the parameter set is assumed. The parameter is fixed in the repeated-sampling coverage probability.

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment with $\mathrm{\Theta }\mathrm{\ne }\mathrm{\varnothing }$, and let $(S,\mathcal{S})$ be a measurable space. A family of measurable maps $q_{\theta }:\mathcal{X}\to S$, one for each $\theta \in \mathrm{\Theta }$, is a pivot if the pushforward law ${\nu }_{\theta }(B):=P_{\theta }(q_{\theta }^{-1}(B)),B\in \mathcal{S}$ is the same probability measure $\nu$ for every $\theta \in \mathrm{\Theta }$. Each section is a random element on $(\mathcal{X},\mathcal{A},P_{\theta })$, so its pushforward is a probability measure by The law of a random element is a probability measure. No measurability in $\theta$ is asserted or needed.

For $S=\mathbb{R}$, a pivot is a parameter-indexed family of measurable real-valued quantities with a common sampling law. The maps $q_{\theta }$ may depend on the unknown parameter. A pivot need not be invertible in the parameter, and its inversion need not be an interval.

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be an experiment with nonempty $\mathrm{\Theta }$. Let $q_{\theta }:\mathcal{X}\to S$ be a pivot with common law $\nu$, and let $B\in \mathcal{S}$. Define $C(x):=\{\theta \in \mathrm{\Theta }:q_{\theta }(x)\in B\}$ If $\nu (B)\ge 1-\alpha$, then $C$ is a confidence set with coverage at least $1-\alpha$. If $\nu (B)=1-\alpha$, its coverage is exactly $1-\alpha$ for every parameter value. For an interval $B=[a,b]$ in $\mathbb{R}$, exactness requires calibration of the actual probability $\nu ([a,b])$; quantile labels alone do not remove endpoint atoms.

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.