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An 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, $D={\sigma }_q^2-{\sigma }_p^2={\sigma }_h^2\ge 0$. An interval $I=[\ell ,u]$ gives a physical width setIt 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 $\hat{D}=-0.030\mathrm{n}{\mathrm{s}}^2$ with known standard deviation $s=0.010\mathrm{n}{\mathrm{s}}^2$. The approximately 95 percent interval $\hat{D}\pm 1.96s=[-0.0496,-0.0104]\mathrm{n}{\mathrm{s}}^2$ 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 $\hat{D}\sim N(D,s^2)$ and $I=[\hat{D}-zs,\hat{D}+zs]$, thenwhere $\Phi$ 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.^[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.^[ref] Neither clipping the estimate nor declaring the response model false follows solely from the sign of one noisy variance difference.

Resolution-limit conditioning in finite-input detector calibration is the sensitivity of a small inferred detector width to uncertainty in two nearly equal measured widths. In finite-input correction in pulse-detector calibration, the nonnegative normalized convolution model givesThe signed relative sensitivities areThey diverge as the measured output width ${\sigma }_q$ approaches the known input width ${\sigma }_p$. Subtracting two well-measured squared widths can therefore be a poorly conditioned inference.

For an original example, ${\sigma }_q=1.00\mathrm{ns}$ and ${\sigma }_p=0.99\mathrm{ns}$ give $D=0.0199\mathrm{n}{\mathrm{s}}^2$ and ${\sigma }_h\approx 0.1411\mathrm{ns}$. The two relative sensitivity coefficients are about $50.25$ and $-49.25$. A small independent fractional width error can produce a much larger fractional response-width error; a common multiplicative timebase error instead partly cancels because the coefficients sum to one. Correlations must therefore be retained.^[ref]

Near the boundary, report uncertainty in the variance difference before transforming it. For a chosen interval $D\in [-0.010,0.050]\mathrm{n}{\mathrm{s}}^2$, imposing the physical constraint $D\ge 0$ gives a detector-width interval $[0,\sqrt{0.050}]\mathrm{ns}$, approximately $[0,0.2236]\mathrm{ns}$. A coverage-preserving detector-width bound from a variance interval uses this monotone square-root transformation on the physical intersection. It need not resemble a symmetric error bar around a point estimate. An interval wholly below zero contains no admissible width under this model; replacing it silently with a zero-width result hides the inconsistency.

NIST characterized its detector/cable/oscilloscope chain with an optical pulse much shorter than the receiver response.^[ref] That design avoids subtracting two nearly equal broad widths. The variance identity still requires nonnegative profiles and finite second moments; arbitrary full widths at half maximum and responses with negative ringing need a different model.