= Boundary probability of an empty detector-width confidence set
{title2=$\sup_{D\geq0}p_{\rm empty}(D)=\Phi(-z)$}
The \b[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 $\widehat D\sim N(D,s^2)$, $D\geq0$, take the central interval $I=[\widehat D-zs,\widehat D+zs]$. The <empty physical confidence set for a detector variance difference> occurs when its upper endpoint is negative:
$$
p_{\rm empty}(D)=\Phi(-z-D/s).
$$
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> $\sqrt{400p(1-p)}\approx3.12$, where $p=\Phi(-1.96)\approx0.025$. 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 $\Phi(-2.96)\approx0.00154$. 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§§6.2–6.3, Expanded Uncertainty]{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.
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