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.

Extremal calibration for nested confidence families obtains simultaneous protection for many candidate bounds by checking their most demanding marginal endpoints. It uses the ordering of a nested family rather than treating every candidate as an unrelated comparison. The simultaneous event then also protects a choice made after observing the data.

Consider the detection model $r=qp$, with upper confidence bounds $r_U(\alpha )$ decreasing in their failure budget $\alpha$ and lower bounds $q_L(\beta )$ increasing in $\beta$. For a prescribed set of candidate allocations $D$, let ${\alpha }_{\ast }$ and ${\beta }_{\ast }$ be the largest respective budgets, with valid bounds defined at those endpoints. Onevery candidate satisfiesThe union bound gives simultaneous coverage at least $1-{\alpha }_{\ast }-{\beta }_{\ast }$. Neither independence of the estimates nor a finite number of candidates is needed for this common-event proof.

For the line segment $\beta ={\gamma }_0-\alpha$, $a\le \alpha \le b$, the endpoints are ${\alpha }_{\ast }=b$ and ${\beta }_{\ast }={\gamma }_0-a$. The protected failure budget is therefore ${\gamma }_0+(b-a)$. The width of the candidate interval consumes part of the simultaneous budget; it cannot be ignored simply because every point on the segment has the same marginal budget sum.

An original exact-tail example has zero detections in $n=37$ and successful detection of all $m=20$ measurement calibration positives. Binomial distribution tail inversion gives $r_U(\alpha )=1-{\alpha }^{1/37}$ and $q_L(\beta )={\beta }^{1/20}$. NIST describes the one-sided tail construction and simultaneous error-budget principle.^[ref] For candidates $(0.01,0.03)$ and $(0.02,0.02)$, the bounds are about 0.13946 and 0.12201. Their extremal endpoint combination $(0.02,0.03)$ gives about 0.11956 and at least 95% coverage. It is a tighter bound with the larger combined budget, and its validity follows directly from the endpoint event.

The construction requires the stated nesting and valid endpoint bounds. A family whose order reverses or whose measurement calibration changes across candidates needs a new argument. If the lower endpoint bound for detection is zero, the prevalence upper bound is one, rather than an undefined division.