Joint probability of empty and strictly positive detector-width intervals ($(N_E,N_P,N_Z)\sim \mathrm{Multinomial}(N;p_E,p_P,p_Z)$)

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.

The negative covariance of empty and positive detector-interval counts arises because a fixed number of measurement intervals is divided into mutually exclusive categories. Let $E_i,P_i$ indicate that interval $i$ is respectively physically empty or strictly positive. The same interval cannot satisfy both, so $E_i{P}_i=0$ andFor $N$ independent repetitions with common probabilities, the counts $N_E={\sum }_i{E}_i$ and $N_P={\sum }_i{P}_i$ therefore haveThe joint probability of empty and strictly positive detector-width intervals gives the underlying multinomial model. The sign describes category-count dependence, not physical antagonism between two detectors.

For an original boundary example, take $N=100$ and $p_E=p_P=0.025$. Each count has variance $100(0.025)(0.975)=2.4375$, while their covariance is $-0.0625$. The total tail-category count has variance $2.4375+2.4375-0.125=4.75$. Treating the two counts as independent would give $4.875$ instead. The covariance is needed even though the individual measurement repetitions are independent.

Conditioning on the total is important. If the number of repetitions is random and independent of the category assignments, the law of total covariance givesAn independent total with a Poisson distribution has equal mean and variance, making this covariance zero; a more variable total can make it positive. This is a different sampling protocol, not a contradiction of the fixed-total formula.

Shared calibration drift is another change of model. Repetitions may then have dependent category indicators, and cross-repetition covariance terms must be added. Overdispersion of detector-interval counts under shared drift can make a simple multinomial analysis inadequate.

NIST's measurement guidance retains common influences in uncertainty propagation and requires assumptions for confidence-level interpretation.^[ref] State whether the repetition total is fixed, how the categories are defined and which calibration influences are shared before interpreting their empirical count covariance.

Overdispersion of detector-interval counts under shared drift is excess count variability caused by a common experimental condition changing the chance of an interval outcome. Let $M$ count empty physical detector-width intervals among $N$ readings. Conditional on a block's calibration state $H$, suppose the readings are independent with empty probability $p(H)$, so $M\mid H\sim \mathrm{Binomial}(N,p(H))$.

Writing $\overline{p}=\mathbb{E}[p(H)]$, the law of total variance givesThe first term is the binomial variance using the mean probability. The second comes from sharing one fluctuating condition across the block. Individual indicators have covariance $\mathrm{Var}(p(H))$ across different readings, even though they are independent after conditioning on $H$.

For an original mixture example, suppose equally likely block conditions have empty probabilities $0.01$ and $0.09$. Then $\overline{p}=0.05$ and $\mathrm{Var}(p(H))=0.0016$. With $N=100$, the expected count is five, but its variance is $4.75+15.84=20.59$, with standard deviation about $4.54$. An independent binomial model with probability $0.05$ would give standard deviation only about $2.18$.

The effective repetition count for detector intervals with shared drift can consequently be much smaller than the number of stored traces. Repeating measurements within one unchanged calibration block does not supply the same information about drift as acquiring independent blocks. Record acquisition grouping and reference reuse before pooling interval categories.

The negative covariance of empty and positive detector-interval counts applies to a fixed-probability multinomial model. Shared drift adds cross-repetition dependence and can change that count covariance as well as each count's variance. In the example, the two probability states are stipulated; a physical model would relate them to the receiver's drift, variance-difference estimator and uncertainty calibration.

NIST retains shared influences in uncertainty propagation and explains that confidence interpretations depend on the assumed distributions.^[ref] Observed overdispersion can motivate checking the block model, but does not by itself identify which physical influence caused it.

The effective repetition count for detector intervals with shared drift is the number of independent repetitions that would give the same variance for an average indicator as the actual dependent repetitions. In overdispersion of detector-interval counts under shared drift, let $X_i$ indicate an empty interval, with common probability $\overline{p}$ satisfying $0<\overline{p}<1$. If the covariance between distinct indicators is $v=\mathrm{Var}(p(H))$, their correlation coefficient isThe parent's law of total variance calculation therefore givesThis is an observable-specific variance equivalence, not a literal count of independent traces or a guarantee of confidence interval coverage. A different statistic or a different dependence pattern may require a different equivalence.

For an original feasible mixture, take $\overline{p}=0.1$ and $\rho =0.1$. These moments can arise when the block probability is $0$ with probability $9/19$ and $0.19$ with probability $10/19$: its variance is $0.009$. With $N=50$, the variance-inflation multiplier is $5.9$ andDoubling the repetitions to $100$ only raises $N_{\mathrm{eff}}$ to about $9.17$. For a fixed positive $\rho$, its large-$N$ limit is $1/\rho$, here $10$.

The experiment must therefore record which traces share a reference or drift state. Acquiring genuinely independent calibration blocks addresses a different limitation from oversampling one block; effective repetition count across independent calibration blocks depends on that grouping. A cluster bootstrap can preserve the observed within-block dependence when blocks are suitable independent resampling units. NIST's uncertainty law explicitly retains shared-input covariance, supporting the need to model reference reuse rather than discard it.^[ref]

Effective repetition count across independent calibration blocks accounts for dependence confined to each acquisition block rather than shared by every stored trace. Suppose block $j$ contains $n_j$ empty-interval indicator random variables, all with common probability $0<p<1$ and within-block correlation coefficient $\rho \ge 0$. Assume different blocks are independent. With $N={\sum }_j{n}_j$, summing within-block covariances givesMatching this to the variance of an average of independent indicators yieldsFor $K$ equal blocks of size $m$, this reduces to $N_{\mathrm{eff}}=Km/[1+(m-1)\rho ]$. The denominator involves the block size, not the total $Km$; the single-block expression in effective repetition count for detector intervals with shared drift must not be applied as though the same drift state affected every block.

For an original comparison, fix $N=120$ and $\rho =0.05$. One block gives $N_{\mathrm{eff}}=120/6.95\approx 17.27$. Six independent blocks of twenty give $N_{\mathrm{eff}}=120/1.95\approx 61.54$. Six very uneven blocks of sizes $(100,4,4,4,4,4)$ instead giveAll three designs store the same number of traces. The difference comes from how often the same fluctuating condition is reused. For fixed $N$, $K$, and positive $\rho$, minimizing ${\sum }_j{n}_j^2$ favors balanced blocks; this motivates optimal allocation of detector repetitions across calibration blocks when block-preparation costs also matter.

Independence must come from the physical acquisition model. Starting another file does not erase a shared clock, reference calibration, or slowly varying environmental influence. NIST's uncertainty law retains shared-input covariance explicitly.^[ref] If drift persists between nominal blocks, those cross-block terms must be added and the displayed formula changes. A cluster bootstrap likewise requires blocks that are suitable independent resampling units.

Optimal allocation of detector repetitions across calibration blocks chooses block sizes using the dependence and acquisition-cost model, not just the desired number of stored traces. For fixed total $N$, number of independent blocks $K$, common success probability, and positive within-block correlation $\rho$, effective repetition count across independent calibration blocks givesMaximizing it is equivalent to minimizing ${\sum }_j{n}_j^2$. If two integer block sizes differ by at least two, transferring one repetition from the larger to the smaller decreases that sum by $2(n_i-n_j-1)>0$. Therefore the optimal integer sizes differ by at most one.

For an original fixed-count example, $N=23$ and $K=5$ give balanced sizes $(5,5,5,4,4)$. At $\rho =0.1$, their effective count is $529/31.4\approx 16.85$. Sizes $(11,3,3,3,3)$ give only $529/36.4\approx 14.53$ under the same assumptions.

More independent blocks can help, but preparing one may consume time. In a continuous planning model let each block cost setup time $h>0$, each repetition cost $\tau >0$, and each block contain $n$ repetitions. With total time $T$, the effective count becomesFor $0<\rho <1$, differentiation gives the interior choice $n_{\ast }=\sqrt{h(1-\rho )/(\tau \rho )}$. Integer block counts, minimum sizes, and hardware restrictions must subsequently be imposed.

For chosen $h=9\mathrm{s}$, $\tau =1\mathrm{s}$, $\rho =0.1$ and $T=180\mathrm{s}$, the continuous optimum $n_{\ast }=9$ is feasible with ten blocks and gives $N_{\mathrm{eff}}=50$. Single-repetition blocks give 18, while two blocks of 81 repetitions also give 18. These are model-based comparisons, not measured apparatus timings.

Unequal block noise or different drift correlations require unequal-noise allocation across calibration blocks. Independence between prepared blocks must also be justified by the reference and environmental model; NIST explicitly retains shared-input covariance.^[ref]

Unequal-noise allocation across calibration blocks distributes repetitions using block-specific variability while keeping the intended estimator fixed. Suppose independent blocks supply means ${\overline{Y}}_j$ and the target estimate is $\hat{\theta }={\sum }_j{w}_j{\overline{Y}}_j$ with specified weights $w_j$. Let one observation in block $j$ have variance ${\sigma }_j^2$ and equicorrelation $0\le {\rho }_j<1$ within that block. ThenThe first sum is a block-specific floor; only the second decreases with repetition count. Changing the estimator weights merely to improve this calculation could change the estimand, so they remain fixed here.

For continuous positive allocations with positive costs $c_j>0$, budget ${\sum }_j{c}_j{n}_j=C$ and $a_j=w_j^2{\sigma }_j^2(1-{\rho }_j)>0$, a Lagrange-multiplier calculation givesFor equal observation costs, repetitions are proportional to $|w_j|{\sigma }_j\sqrt{1-{\rho }_j}$, not to the variance itself. Minimum counts and integer rounding must subsequently respect the actual acquisition constraints.

For an original two-block example, take $w_1=w_2=1/2$, equal costs, $C=100$, independent readings within each block, and standard deviations $1$ and $3$ in the output unit. The optimal counts are $25$ and $75$, giving variance $0.04$ instead of $0.05$ from fifty each. If both within-block correlations are instead $0.1$, the same proportions give variance $0.286$, while equal counts give $0.295$. The common-error floor $0.25$ now dominates and cannot be eliminated by taking more observations inside those same blocks.

This extends optimal allocation of detector repetitions across calibration blocks, whose balanced solution used homogeneous noise. For binary detector indicators, the per-reading variance is $p_j(1-p_j)$; differing probabilities require a clearly specified weighted target. The planning variances and correlations must come from an appropriate measurement calibration model, not merely from counting files as independent blocks. NIST's uncertainty propagation retains shared-input covariance for exactly this reason.^[ref]