= Reciprocal-estimation bias in randomized atom inclusion
<Reciprocal-estimation bias in randomized atom inclusion> quantifies the nonlinear plug-in loss beyond the sign supplied by <Jensen's inequality>. Keep the independent-calibration model of <atom-probability error in a randomized endpoint rule>: true atom probability $p>0$, unbiased estimate $\widehat p=p+\varepsilon$, and included probability $C=b-ap/\widehat p$, with the upper endpoint calibrated to actual probability $b$. Assume $0<a<b<1$, $\widehat p\ge a$, and $E\varepsilon=0$.
The exact algebraic identity
$$
\frac1{p+\varepsilon}=\frac1p-\frac{\varepsilon}{p^2}
+\frac{\varepsilon^2}{p^2(p+\varepsilon)}
$$
gives
$$
EC-(b-a)=-\frac ap E\!\left(\frac{\varepsilon^2}{p+\varepsilon}\right).
$$
Thus the loss depends on squared errors weighted by the reciprocal estimated probability. The second-order <delta method> approximation is $-a\operatorname{Var}(\widehat p)/p^2$, but the exact remainder shows why small estimated probabilities deserve particular attention.
If $|\varepsilon|\le\eta p$ almost surely for $0<\eta<1$, a nonasymptotic bound is
$$
-\frac{a\operatorname{Var}(\widehat p)}{p^2(1-\eta)}
\le EC-(b-a)\le
-\frac{a\operatorname{Var}(\widehat p)}{p^2(1+\eta)}.
$$
For an original small-error example, let $p=0.4$, let <measurement calibration> return $0.36$ or $0.44$ equally often, and take $a=0.025$, $b=0.975$. Here $\eta=0.1$ and the <variance> is $0.0016$. The exact included probability is about $0.9497475$; the second-order approximation is $0.94975$, while the bounded-error calculation puts the coverage loss between about $0.0002273$ and $0.0002778$.
The bounded-relative-error assumption can fail for a finite binomial <measurement calibration>: an empirical atom fraction can be zero with positive probability. Such <near-zero probability estimates in reciprocal plug-in rules> require an explicitly defined guard or different <measurement calibration> model, not division by zero. The auxiliary endpoint decision must remain independent of the future duration and <measurement calibration> as stipulated. The documented uniform random-variates construction supplies that auxiliary randomness; it does not validate the atom estimate.<SciPy 1.16.2 documentation>[scipy.stats.uniform, support and rvs]{ref}
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