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Continuous fixed-size Neyman allocation minimizes stratified variance

Statement

Use the notation of Stratified sampling designs and the stratified total estimator and Unbiasedness and variance of the stratified total estimator. Let $J$ be a nonempty set of noncertainty strata whose allocations vary, and suppose $S_{h} > 0$ for each $h \in J$ , where $S_{h} : = \sqrt{S_{h}^{2}}$ is the positive population standard deviation. Hold all other stratum allocations fixed. For real $x_{h} > 0$ with $\sum_{h \in J} x_{h} = n^{⋆} > 0$ , consider the uncapped continuous relaxation of the variance expression $V_{0} + \sum_{h \in J} N_{h}^{2} (1 - \frac{x_{h}}{N_{h}}) \frac{S_{h}^{2}}{x_{h}},$ where $V_{0}$ collects the fixed strata's variance terms. Its unique minimum over this positive simplex is attained at $x_{h} = n^{⋆} \frac{N_{h} S_{h}}{\sum_{k \in J} N_{k} S_{k}} (h \in J) .$ This is the continuous Neyman allocation. The relaxation imposes no integer, capacity, lower-bound, or minimum-sample constraints; if the displayed allocation violates a design constraint, it is not itself an admissible stratified SRSWOR design.

Facts & Assumptions

Given: A stratified SRSWOR finite-population design; a nonempty set $J$ of variable noncertainty strata with $S_{h} > 0$ ; fixed allocations outside $J$ ; and a positive real budget $n^{⋆}$ for $J$ .

[F1]

Stratified sampling designs and the stratified total estimator: For a noncertainty stratum, $N_{h} > 1$ and $S_{h}^{2} = (N_{h} - 1)^{- 1} \sum_{i \in U_{h}} (y_{i} - {\overset{ˉ}{y}}_{h})^{2}$ .

[F2]

Unbiasedness and variance of the stratified total estimator: A noncertainty stratum's variance contribution is $N_{h}^{2} (1 - n_{h} / N_{h}) S_{h}^{2} / n_{h}$ ; certainty and fixed strata contribute terms that do not vary with $x_{h}$ .

Proof

1.1givenF1F2algebra

Put $a_{h} = N_{h} S_{h} > 0$ . For $h \in J$ , [F2]'s variable term with $n_{h} = x_{h}$ is $N_{h}^{2} (1 - x_{h} / N_{h}) S_{h}^{2} / x_{h} = a_{h}^{2} / x_{h} - N_{h} S_{h}^{2}$ . The subtracted term is constant, so the relaxed objective is minimized by minimizing $\sum_{h \in J} a_{h}^{2} / x_{h}$ subject to $\sum_{h \in J} x_{h} = n^{⋆}$ .

2.1givenstep 1.1algebra

Set $p_{h} = a_{h} / \sqrt{x_{h}}$ and $q_{h} = \sqrt{x_{h}}$ . Expanding and grouping unordered pairs gives $(\sum_{h} p_{h}^{2}) (\sum_{h} q_{h}^{2}) - (\sum_{h} p_{h} q_{h})^{2} = \sum_{h < k} (p_{h}^{2} q_{k}^{2} + p_{k}^{2} q_{h}^{2} - 2 p_{h} p_{k} q_{h} q_{k}) = \sum_{h < k} (p_{h} q_{k} - p_{k} q_{h})^{2} \geq 0$ . Since $\sum_{h} q_{h}^{2} = n^{⋆}$ and $\sum_{h} p_{h} q_{h} = \sum_{h} a_{h}$ , we get $\sum_{h} a_{h}^{2} / x_{h} \geq (\sum_{h} a_{h})^{2} / n^{⋆}$ , with equality exactly when $a_{h} / x_{h}$ is constant over $J$ .

3.1givenstep 1.1step 2.1algebra∎

Define $x_{h}^{⋆} = n^{⋆} a_{h} / (\sum_{k \in J} a_{k})$ . Positivity gives $x_{h}^{⋆} > 0$ , and summing gives $\sum_{h} x_{h}^{⋆} = n^{⋆}$ . This allocation satisfies step 2.1's equality condition and attains its lower bound; the equality condition uniquely fixes this normalization. By step 1.1 it uniquely minimizes the relaxed variance.

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Sources

Shalabh, Sampling Theory, MTH 417 lecture notes, Chapter 4: Stratified Sampling, PDF pp. 8–9 and 12–14 (standard reference, not scraped)
Robin Arnold, STAT392 Sample Surveys, Chapter 8: Stratified Sampling, §8.8 (standard reference, not scraped)
Tommy Wright, A Simple Method of Exact Optimal Sample Allocation under Stratification with Any Mixed Constraint Patterns, U.S. Census Bureau, §2.1 (standard reference, not scraped)