 Boundary probability of an empty detector-width confidence set

The boundary probability of an empty detector-width confidence set is the chance that a valid uncertainty procedure yields no admissible nonnegative response width when the true variance difference is at its boundary. Under the explicit model $\hat{D}\sim N(D,s^2)$, $D\ge 0$, take the central interval $I=[\hat{D}-zs,\hat{D}+zs]$. The empty physical confidence set for a detector variance difference occurs when its upper endpoint is negative:This probability decreases with $D$, so the largest value over the physical parameter range occurs at $D=0$. If $z$ is the usual two-sided $1-\alpha$ normal critical value, the boundary probability is $\alpha /2$.

At that boundary the other way to miss zero is a strictly positive lower endpoint. The two tail events are disjoint and each has probability $\alpha /2$; the remaining probability $1-\alpha$ produces a confidence set containing zero. The joint probability of empty and strictly positive detector-width intervals therefore accounts for the full two-sided noncoverage, not just the empty cases.

For an original repeated-measurement calculation, use $z=1.96$ and 400 independent measurements of a stipulated zero response-variance difference. The expected number of empty physical sets is approximately ten, with standard deviation $\sqrt{400p(1-p)}\approx 3.12$, where $p=\Phi (-1.96)\approx 0.025$. The count follows a binomial distribution only under the stated independent, common-probability model. Roughly ten additional intervals are expected to lie wholly above zero. Their existence is compatible with approximately 95 percent coverage at the boundary.

For an interior example $D=s$, the empty-set probability falls to $\Phi (-2.96)\approx 0.00154$. Empty results are thus most common near the resolution boundary, where subtracting similar measured variances is already delicate.

NIST explains that interpreting expanded uncertainty as a confidence interval requires distributional assumptions.^[ref] These probabilities assume an actual normal model with known $s$; they are not inferred merely from an error bar labelled standard uncertainty. Correlation, systematic offsets or a misspecified detector response can change the repeated-measurement behavior. Retaining empty results in the record allows that behavior to be assessed rather than hiding it by replacing them with zero-width estimates.