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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.

The joint probability of empty and strictly positive detector-width intervals must distinguish outcomes of one measurement from counts across repeated measurements. For one normal variance-difference estimate $\hat{D}\sim N(D,s^2)$, let $I=[\hat{D}-zs,\hat{D}+zs]$. An empty physical width set requires $\hat{D}+zs<0$; a strictly positive interval requires $\hat{D}-zs>0$. Those events are disjoint for that same interval, so their intersection has probability zero.

Writing $d=D/s$ and $\Phi$ for the standard normal cumulative distribution function, the three category probabilities areThe last category is an interval containing zero. At $D=0$ and a central $1-\alpha$ interval, these become $(\alpha /2,\alpha /2,1-\alpha )$, as in boundary probability of an empty detector-width confidence set. For interior $D>0$, the categories describe the relation to zero; they are not themselves the coverage categories for the true positive value.

For $N$ independent repetitions with the same probabilities, the counts $(N_E,N_P,N_Z)$ have a multinomial distribution. For nonnegative integers $e,p,k$ summing to $N$,Although the measurement repetitions are independent, their category counts are not. An empty and a positive indicator cannot both equal one in the same repetition, givingThis negative covariance of empty and positive detector-interval counts follows from the fixed total count.

For an original example, take 20 independent boundary measurements with exact central 95 percent normal intervals. Then $p_E=p_P=0.025$. Each count has mean $0.5$, and their covariance is $-0.0125$. The probability of exactly one empty and one strictly positive interval is $20\times 19\times 0.025^2\times 0.95^{18}$, about $0.094$. It is not zero, unlike the intersection of those two outcomes for a single interval.

NIST's uncertainty guidance emphasizes the assumptions needed to assign confidence levels.^[ref] The joint-count calculation above additionally assumes independent repetitions with a common normal model; shared calibration errors can invalidate that independence even when each marginal interval is computed in the same way.