= Coverage-preserving detector-width bound from a variance interval
{title2=$J=\sqrt{I\cap[0,\infty)}$}
A \b[coverage-preserving detector-width bound from a <variance> interval] transforms uncertainty in a nonnegative response <variance> without imposing a symmetric width error bar. Let $D=\sigma_h^2\geq0$ and let a random interval $I(X)$ have stated frequentist coverage for $D$. Define the width confidence set by
$$
J(X)=\{\sqrt d:d\in I(X)\cap[0,\infty)\}.
$$
For the true width $\sigma_h\geq0$,
$$
\{\sigma_h\in J(X)\}=\{D\in I(X)\}.
$$
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 $\widehat D$ has a <normal distribution> with mean $D$ and known <standard deviation> $s=0.025\,\mathrm{ns^2}$. A reading $\widehat D=0.040\,\mathrm{ns^2}$ gives an approximately 95 percent interval $\widehat D\pm1.96s=[-0.009,0.089]\,\mathrm{ns^2}$. 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.<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} 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.
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