Exact selective confidence set for a parameter functional

Let $\mathsf{E}=((\mathcal{Y},\mathcal{G}),\mathcal{M},(Q_{\mu }{)}_{\mu \in \mathcal{M}})$ be a statistical experiment (Statistical experiments) and let $\tau :\mathcal{M}\to \mathrm{\Lambda }$ be a parameter functional. For $0\le \alpha \le 1$, a set-valued procedure $C:\mathcal{Y}\to \mathcal{P}(\mathrm{\Lambda })$ is a selective confidence set with coverage at least $1-\alpha$ if, for every $\mu \in \mathcal{M}$, its membership section $\{y:\tau (\mu )\in C(y)\}$ belongs to $\mathcal{G}$ and $Q_{\mu }\{y:\tau (\mu )\in C(y)\}\ge 1-\alpha \mathrm{.}$ It is exact if equality holds for every $\mu$. Here the laws $Q_{\mu }$ are the laws after conditioning on the specified selection event. When $\mathcal{M}\mathrm{\ne }\mathrm{\varnothing }$, the pullback $D(y)=\{\mu \in \mathcal{M}:\tau (\mu )\in C(y)\}$ is a confidence-set procedure as in Confidence sets and coverage, with the same membership sections and coverage at each $\mu$. When $\mathcal{M}=\mathrm{\varnothing }$, the stated coverage and exactness requirements are vacuous. No measurable structure on $\mathcal{M}$ or $\mathrm{\Lambda }$ is imposed.