Unequal-noise allocation across calibration blocks

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

θ = \sum_{j} w_{j} \overline{Y}_{j}

with specified weights

w_{j}

. Let one observation in block

j

σ_{j}^{2}

and equicorrelation

0 \leq ρ_{j} < 1

within that block. Then

Var (θ) = \sum_{j} w_{j}^{2} σ_{j}^{2} ρ_{j} + \sum_{j} \frac{w _{j}^{2} σ _{j}^{2} ( 1 - ρ _{j} )}{n _{j}} .

The 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} σ_{j}^{2} (1 - ρ_{j}) > 0

n_{j} = \frac{C a _{j} / c _{j}}{\sum _{k} a _{k} c _{k}}, min \sum_{j} \frac{a _{j}}{n _{j}} = \frac{( \sum _{j} a _{j} c _{j} ) ^{2}}{C} .

For equal observation costs, repetitions are proportional to

∣ w_{j} ∣ σ_{j} 1 - ρ_{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

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]

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