 Uniform coverage over confidence-allocation choices

Uniform coverage over confidence-allocation choices means that one probability statement protects every allocation in a specified family simultaneously. For upper bounds $B_{\lambda }(X)$ on a parameter $\theta$, the required statement isIt implies coverage for the minimum selected after observing the data. Separate statements ${\mathrm{Pr}}_{\theta }\{\theta \le B_{\lambda }(X)\}\ge 1-\gamma$ for each fixed $\lambda$ do not imply the simultaneous statement. The difference is the order of the probability and the requirement to cover every choice.

For a finite set of $M$ candidates, assigning each a failure budget $\gamma /M$ and applying the union bound provides uniform protection. This is the simultaneous-coverage principle behind the Bonferroni correction.^[ref] A continuum requires a family-level argument rather than multiplying one finite-candidate budget by an undefined number of choices.

An original construction exploits nested one-sided bounds in the detection model $r=qp$. Suppose $r_U(\alpha )$ decreases as its failure budget $\alpha$ increases, and $q_L(\beta )$ increases as $\beta$ increases. Choose allocations $\alpha \in [a,b]$, $\beta ={\gamma }_0-\alpha$, with $0<a\le b<{\gamma }_0$. On the common eventevery candidate ratio $r_U(\alpha )/q_L({\gamma }_0-\alpha )$ bounds $p$. The union bound controls this common event's failure probability byThis extremal calibration for nested confidence families protects the entire interval of allocations using just its most demanding marginal endpoints.

For example, take ${\gamma }_0=0.04$, $a=0.01$ and $b=0.02$. Each fixed allocation has at least 96% coverage, while the common-event argument guarantees at least 95% coverage for every allocation simultaneously. Selecting the smallest bound within that prespecified interval is therefore protected at the latter level, without assuming independence of the marginal bounds.

The proof depends on the nesting, valid marginal bounds and a specified family. Enlarging the family after inspecting the observations needs corresponding protection. Where a lower detection bound can be zero, the prevalence upper bound must revert to the uninformative value one rather than divide by zero.