 Error-budget allocation between prevalence and detection calibration

Error-budget allocation between prevalence and detection calibration assigns the allowed failure probability of a joint confidence bound between the specimen survey and its detection-calibration experiment. Under the model $r=qp$, an upper bound $r_U$ with failure probability at most $\alpha$ and a positive lower bound $q_L$ with failure probability at most $\beta$ give $p_U=\min \{1,r_U/q_L\}$. The union bound guarantees coverage at least $1-\alpha -\beta$. A total budget $\gamma$ therefore permits any prespecified positive allocation satisfying $\alpha +\beta =\gamma$; equal allocation is only one choice.

For independent specimens with zero detections among $n$ and all successful detections among $m$ known-positive calibration specimens, exact binomial distribution tail inversion givesThe tail construction is the one-sided method described by NIST.^[ref] For this anticipated outcome pattern, the uncapped bound becomesMaking $\alpha$ larger tightens the numerator but leaves less failure budget for calibration, which lowers $q_L$ and weakens the denominator. The two effects compete.

An original planning example sets $n=37$, $m=20$ and $\gamma =0.05$. Equal allocation gives $U(0.025)\approx 0.11411$. A fixed alternative allocation $\alpha =0.043$, $\beta =0.007$ gives approximately $0.10448$, still with at least 95% joint coverage. For a differentiable interior optimum of this specific objective,The example is minimized near $\alpha =0.04295$. It is not a universal optimal split for other counts or detection models.

Prespecified versus data-selected confidence allocations is crucial. Choosing the most favorable allocation after inspecting many outcomes does not inherit this fixed-allocation coverage proof. Design the allocation before observation, or use a construction whose validity includes the selection rule. Calibration must also address the same detection difficulty as the survey; redistributing failure probabilities does not repair an inappropriate calibration population.