Atom-probability error in a randomized endpoint rule

Atom-probability error in a randomized endpoint rule is the change in included duration-model probability caused by using an inaccurate zero-atom probability in the randomization. Let $0<a<b<1$, and hold the positive upper endpoint $u$ fixed at an actual cumulative distribution function value $F(u)=b$. If the true zero probability is $p_0$ but the rule uses a positive ${\hat{p}}_0\ge a$, its independent endpoint-inclusion probability is $r=1-a/{\hat{p}}_0$. Conditional on that chosen value,This is the atom term isolated in distribution sensitivity of randomized zero-endpoint probabilities, with upper-endpoint error deliberately removed. Underestimating $p_0$ reduces inclusion; overestimating it increases inclusion. The independent auxiliary draw can be generated from a uniform distribution.^[ref]

If ${\hat{p}}_0$ comes from an independent calibration, conditional probabilities must also be averaged over that calibration. For an original example, take $p_0=0.20$, $a=0.025$, and $b=0.975$. Suppose the calibration returns $0.10$ or $0.30$ with equal probability. Its estimate is unbiased, yet the resulting included probabilities are $0.925$ and $0.95833$, averaging about $0.94167$ rather than $0.95$.

More generally, for an unbiased positive estimate in the valid rule range, Jensen's inequality givesThis reciprocal-estimation bias in randomized atom inclusion comes from a nonlinear plug-in rule, even with independent calibration. It does not identify the size of the error without a calibration model. Values below $a$, including zero, require an explicitly defined clipped or guarded rule; the displayed inverse formula cannot simply be used there.

If a validated upper bound $p_U\ge p_0$ is available and $p_U\ge a$, choosing $r=1-a/p_U$ instead guarantees $C\ge b-a$ under the fixed-upper-endpoint and independence assumptions. It may be conservative rather than exact. Uncertainty in $F(u)$ remains a separate error term, and these are probabilities for a duration distribution, not a removal of the physical zero-duration atom.

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 $\hat{p}=p+\epsilon$, and included probability $C=b-ap/\hat{p}$, with the upper endpoint calibrated to actual probability $b$. Assume $0<a<b<1$, $\hat{p}\ge a$, and $E\epsilon =0$.

The exact algebraic identitygivesThus the loss depends on squared errors weighted by the reciprocal estimated probability. The second-order delta method approximation is $-a\mathrm{Var}(\hat{p})/p^2$, but the exact remainder shows why small estimated probabilities deserve particular attention.

If $|\epsilon |\le \eta p$ almost surely for $0<\eta <1$, a nonasymptotic bound isFor 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.^[ref]

Near-zero probability estimates in reciprocal plug-in rules can make a nominal randomization undefined even when the underlying probability estimate is unbiased. For an independent measurement calibration count $J\sim \mathrm{Binomial}(M,p)$, the empirical atom fraction $\hat{p}=J/M$ has $E\hat{p}=p$, butThus its reciprocal is not a finite statistic on every measurement calibration outcome. The binomial probability mass function supplies this zero-count probability directly.^[ref]

In reciprocal-estimation bias in randomized atom inclusion, a fixed upper endpoint contributes actual probability $b$ and the auxiliary rule uses $r=1-a/\hat{p}$. Valid probabilities require a separate rule when $\hat{p}<a$, including zero. One possible numerical guard is ${\hat{p}}_{ϵ}=\max (J/M,ϵ)$ with $a\le ϵ\le 1$. It makes $r=1-a/{\hat{p}}_{ϵ}$ defined and between zero and one. It does not preserve the unbiasedness or the coverage properties of the unguarded formula.

For an original finite measurement calibration example, take $M=20$, $p=0.02$, $a=0.005$, $b=0.995$, and $ϵ=0.01$. A zero count occurs with probability about $0.6676$. On that outcome the guarded rule includes zero with probability $0.5$, and its included duration-model probability is $b-ap/ϵ=0.985$, below the nominal $b-a=0.99$. Averaging over every measurement calibration outcome givesThe calculation assumes the future duration, measurement calibration sample, and auxiliary random draw have the stipulated independence, and the upper-endpoint probability is exactly $b$.

Such finite-calibration guards for reciprocal endpoint randomization solve a domain problem, not a coverage proof. A validated upper probability bound can provide a different conservative construction under the parent's assumptions, but an arbitrary floor is not automatically such a bound. Calibration size, lower-tail events and the actual guarded rule must all enter the final joint-probability calculation.

Finite-calibration guards for reciprocal endpoint randomization must specify both a valid endpoint decision and how calibration uncertainty enters its probability guarantee. An arbitrary positive floor avoids division by zero but does not establish nominal inclusion, as the example in near-zero probability estimates in reciprocal plug-in rules demonstrates.

One finite-sample construction uses an upper confidence bound $U(J)$ for the atom probability $p$, based on independent calibration count $J\sim \mathrm{Binomial}(M,p)$. Supposewhere $0<\gamma <1$ is a calibration failure budget. An exact one-sided binomial bound can be obtained by inversion: for $j<M$, choose $U(j)$ satisfying ${\mathrm{Pr}}_{U(j)}(J\le j)=\gamma$, and put $U(M)=1$. This is a confidence statement over calibration samples, not a posterior probability for a fixed $p$ after observing one count. The pinned reference documents the exact binomial proportion-interval method.^[ref]

For a fixed upper duration endpoint with actual cumulative probability $b$, setThe rule is defined and $0\le r\le 1$. On the event $p\le U(J)$, its conditional included probability is $b-ap/q(J)\ge b-a$. This calculation also requires independence of the calibration, future duration and auxiliary endpoint draw as stipulated by the parent model.

For an original zero-count example, take $M=20$ and $\gamma =0.01$. Since ${\mathrm{Pr}}_p(J=0)=(1-p{)}^{20}$, inversion gives $U(0)=1-0.01^{1/20}\approx 0.20567$. With $a=0.005$, $b=0.995$, and stipulated true $p=0.02$, the zero-count conditional included probability is approximately $0.99451$. The bound is much larger than the raw estimate zero, reflecting the uncertainty of a small calibration sample rather than inventing a measured atom fraction.

The calibration confidence level is not free. Since included probabilities are nonnegative even on the failure event, a general joint guarantee from this argument is onlyAt conditional target $0.99$ and calibration confidence $0.99$, this lower guarantee is $0.9801$, not $0.99$. To certify an unconditional target $t$ using this argument, the conditional target must be at least $t/(1-\gamma )$, provided it can fit below $b$, or the failure budget must be reduced. Any upper-endpoint model error also needs separate accounting.