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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.

Finite-input correction in pulse-detector calibration removes the known input duration from an inferred receiver width, while propagating uncertainty in both widths. Under the nonnegative, normalized, finite-variance convolution model of known-input calibration of a pulse detector,where $p$ is the known input profile, $q$ the measured output and $h$ the receiver's impulse response. The subtraction concerns temporal variances, not arbitrary reported full widths.

Differentiating on the positive-width branch givesFirst-order propagation of standard uncertainty therefore yieldsThe covariance matters when both widths use the same timebase or fitted reference.^[ref]

An original near-resolution example has ${\sigma }_p=0.95\mathrm{ns}$ and ${\sigma }_q=1.00\mathrm{ns}$, each with independent standard uncertainty $0.01\mathrm{ns}$. The corrected receiver width is $0.31225\mathrm{ns}$ and its propagated standard uncertainty is about $0.04417\mathrm{ns}$, or $14.1\%$. Width measurements individually precise to about one percent can thus give a much less precise difference-derived width.

NIST's primary bandwidth-correction procedure used roughly 120 ps optical inputs and recorded the detector, cable and oscilloscope as one response chain at different gains and signal levels.^[ref] A sufficiently short characterized input reduces this resolution-limit conditioning in finite-input detector calibration. Near a zero variance difference, linear uncertainty propagation may fail; a nonpositive measured difference does not represent a negative physical detector width. Report an interval or limit under the stated model rather than silently replacing the difference by zero. Negative ringing also violates the nonnegative-profile assumption behind the variance subtraction.