 Distribution sensitivity of randomized zero-endpoint probabilities

Distribution sensitivity of randomized zero-endpoint probabilities quantifies how the probability equality in randomized endpoint inclusion at a zero-duration atom changes when its assumed duration law is wrong. Let the nominal law have zero-atom probability ${\tilde{p}}_0$, choose a positive upper endpoint $\tilde{u}$ with nominal cumulative distribution function $\tilde{F}(\tilde{u})=b$, and put $r=1-a/{\tilde{p}}_0$, with $0<a\le {\tilde{p}}_0<b<1$. Use the parent's independently randomized decision to include zero.

If the actual zero probability is $p_0$ and the actual cumulative distribution function is $F$, the included probability isSubtracting the target $b-a$ separates the two model errors:The first term is upper-endpoint probability error; the second is atom-probability error in a randomized endpoint rule. This identity is an algebraic consequence of independence of the auxiliary random draw, not a normal-approximation result. The documented uniform distribution random-variates construction supplies that auxiliary draw; normal cdf/ppf routines can evaluate a specified clipped-normal duration model.^[ref]

For an original misspecification example, choose $a=0.025$, $b=0.975$, and ${\tilde{p}}_0=0.20$, so $r=0.875$. Suppose instead $p_0=0.30$ and $F(\tilde{u})=0.970$. Thenwhich is $0.0175$ below $0.95$: $0.005$ comes from the upper endpoint and $0.0125$ from the atom. These are stipulated distributions, not measured failure rates.

More generally, bounds $|F(\tilde{u})-b|\le {ϵ}_F$ and $|p_0-{\tilde{p}}_0|\le {ϵ}_p$ imply $|C-(b-a)|\le {ϵ}_F+(a/{\tilde{p}}_0){ϵ}_p$. This propagates validated model-error bounds only. Estimating the nominal law from the same observation used for the interval can add dependence; the unconditional fixed-rule identity then requires an appropriate joint or conditional argument.