Exact selective tests and confidence sets from the pivot

For any $\mu$ in the conditional experiment, [F1] gives $q_{{\eta }^T\mu }(Y)\sim \mathrm{Uniform}(0,1)$ under $Q_{\mu }$. For $0\le u\le 1$, the event $2\min \{q,1-q\}\le u$ is the union of the two tails $q\le u\mathrm{/}2$ and $q\ge 1-u\mathrm{/}2$, whose uniform probabilities sum to $u$ (their intersection has measure zero). Thus, if $H_0(\vartheta )\mathrm{\ne }\mathrm{\varnothing }$, $p_{\vartheta }$ is uniform under every null law and is measurable by the pivot formula; [F2] makes it a valid p-value.

If $H_0(\vartheta )\mathrm{\ne }\mathrm{\varnothing }$ and the alternative is nonempty, [F3] applied to $p_{\vartheta }$ gives conditional size at most $\alpha$. By [F4] size is the supremum of the null rejection probabilities; since every null law has the exact uniform p-value distribution from step 1.1, $Q_{\mu }(p_{\vartheta }\le \alpha )=\alpha$ for every null $\mu$, so the size is exactly $\alpha$.

By [F1], [F6], and [F8], $q_{\mu }(Y):=q_{{\eta }^T\mu }(Y)$ is a pivot in the conditional experiment. Its full-parameter inversion is $C_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(Y)=\{{\mu }^{\mathrm{\prime }}\in {\mathrm{\Theta }}_S:q_{{\eta }^T{\mu }^{\mathrm{\prime }}}(Y)\in [\alpha \mathrm{/}2,1-\alpha \mathrm{/}2]\}$. The acceptance interval has uniform probability $1-\alpha$, so [F7] gives exact coverage for $C_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}$. Its image under ${\mu }^{\mathrm{\prime }}\mapsto {\eta }^T{\mu }^{\mathrm{\prime }}$ is exactly $C_{\eta }(Y)$ by the definition of ${\mathrm{\Theta }}_{\eta }$. For each fixed true $\mu$, membership ${\eta }^T\mu \in C_{\eta }(Y)$ is precisely $\alpha \mathrm{/}2\le q_{{\eta }^T\mu }(Y)\le 1-\alpha \mathrm{/}2$, which has $Q_{\mu }$-probability $1-\alpha$ by [F1]. The membership section is measurable by the Borel pivot formula, so [F5] applies.