= Error-budget allocation between prevalence and detection calibration
\b[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 gives
$$
r_U=1-\alpha^{1/n},\qquad q_L=\beta^{1/m}.
$$
The tail construction is the one-sided method described by NIST.<NIST and SEMATECH e-Handbook of Statistical Methods>[§7.2.4.1, One-sided confidence intervals and Exact Intervals for Small Numbers of Failures]{ref} For this anticipated outcome pattern, the uncapped bound becomes
$$
U(\alpha)=\frac{1-\alpha^{1/n}}{(\gamma-\alpha)^{1/m}},\qquad 0<\alpha<\gamma.
$$
Making $\alpha$ 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 $U(0.025)\approx0.11411$. 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,
$$
\frac{d\log U}{d\alpha}
=-\frac{\alpha^{1/n-1}}{n(1-\alpha^{1/n})}
+\frac{1}{m(\gamma-\alpha)}=0.
$$
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.
Back to article page
