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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.

A coverage-preserving detector-width bound from a variance interval transforms uncertainty in a nonnegative response variance without imposing a symmetric width error bar. Let $D={\sigma }_h^2\ge 0$ and let a random interval $I(X)$ have stated frequentist coverage for $D$. Define the width confidence set byFor the true width ${\sigma }_h\ge 0$,The events are identical, so their coverage probabilities are identical. The transformation does not create coverage that the original interval lacked. Resolution-limit conditioning in finite-input detector calibration explains why working with the variance difference first is useful near a barely resolved detector response.

For an original explicit measurement model, suppose $\hat{D}$ has a normal distribution with mean $D$ and known standard deviation $s=0.025\mathrm{n}{\mathrm{s}}^2$. A reading $\hat{D}=0.040\mathrm{n}{\mathrm{s}}^2$ gives an approximately 95 percent interval $\hat{D}\pm 1.96s=[-0.009,0.089]\mathrm{n}{\mathrm{s}}^2$. Intersecting with the physical range and taking square roots gives $J=[0,0.2983]\mathrm{ns}$. This is an assumed normal model for measuring $D$, not a claim that any difference of fitted squared widths is automatically normal.

If the entire variance interval is negative, the result is an empty physical confidence set for a detector variance difference. Do not replace its negative upper endpoint by zero and report an apparently exact zero width: that changes the confidence-set construction. An empty realization signals disagreement between that interval and the physical parameter range; it can occur even in a valid procedure on an occasional sample that misses the true value.

NIST distinguishes a standard uncertainty from an interval with a justified confidence level and explains the distributional assumptions behind a coverage factor near two.^[ref] The normal interval above needs its stated model; the monotone transformation proof itself does not. Application to a detector additionally requires the nonnegative-profile convolution assumptions of finite-input correction in pulse-detector calibration; a correct statistical transformation cannot repair an invalid physical variance identity.