= 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 $\widehat\theta=\sum_j w_j\overline Y_j$ 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. Then
$$
\operatorname{Var}(\widehat\theta)=
\sum_jw_j^2\sigma_j^2\rho_j+
\sum_j\frac{w_j^2\sigma_j^2(1-\rho_j)}{n_j}.
$$
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 $\sum_j c_jn_j=C$ and $a_j=w_j^2\sigma_j^2(1-\rho_j)>0$, a Lagrange-multiplier calculation gives
$$
n_j=\frac{C\sqrt{a_j/c_j}}{\sum_k\sqrt{a_kc_k}},\qquad
\min\sum_j\frac{a_j}{n_j}=\frac{(\sum_j\sqrt{a_jc_j})^2}{C}.
$$
For 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
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