= Empty physical confidence set for a detector variance difference
{title2=$J=\varnothing\iff\sup I<0$}
An \b[empty physical <confidence set> for a detector <variance> difference] occurs when the entire reported interval for the inferred response <variance> lies below its admissible range. In the nonnegative-profile detector model, $D=\sigma_q^2-\sigma_p^2=\sigma_h^2\geq0$. An interval $I=[\ell,u]$ gives a physical width set
$$
J=\{\sqrt d:d\in I\cap[0,\infty)\}.
$$
It is empty exactly when $u<0$. The <coverage-preserving detector-width bound from a variance interval> proves why this intersection and transformation retain the original coverage event.
For an original reading, suppose an explicitly normal measurement model for $D$ gives $\widehat D=-0.030\,\mathrm{ns^2}$ with known <standard deviation> $s=0.010\,\mathrm{ns^2}$. The approximately 95 percent interval $\widehat D\pm1.96s=[-0.0496,-0.0104]\,\mathrm{ns^2}$ contains no nonnegative <variance>. This is disagreement between that interval and the physical model, not a negative physical detector width or an exact measurement of zero width.
Empty sets can occur occasionally even when the model is correct. If $\widehat D\sim N(D,s^2)$ and $I=[\widehat D-zs,\widehat D+zs]$, then
$$
\Pr(J=\varnothing)=\Pr(\widehat D+zs<0)=\Phi(-z-D/s),
$$
where $\Phi$ is the standard normal cumulative distribution function. At the physical boundary $D=0$ and $z=1.96$, the probability is about $2.5\%$; at $D=s$ it is about $0.154\%$. The <boundary probability of an empty detector-width confidence set> explains why one empty realization is not a logical disproof of the detector model.
Replacing an empty set by the point zero changes the procedure. It must not be presented as though the original interval precisely determined that point. NIST likewise distinguishes an uncertainty interval from an automatically justified confidence level, emphasizing the distributional assumptions behind coverage factors.<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}
In practice, examine the width estimates, their correlations and the physical <convolution> assumptions, while retaining the actual negative interval in the measurement record. NIST's short-input receiver-response measurement illustrates how an independently characterized input helps resolve detector broadening.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37, Figure C1]{ref} Neither clipping the estimate nor declaring the response model false follows solely from the sign of one noisy <variance> difference.
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