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Continuous Neyman allocation may require constrained rounding

Example

Take two strata of sizes $2$ and $10$ . In the first, let the values be $(- 1 / \sqrt{2}, 1 / \sqrt{2})$ ; in the second, take five copies each of $- 9 / \sqrt{10}$ and $9 / \sqrt{10}$ . With total sample size six, the continuous Neyman allocation rounds to zero units in the first stratum under nearest- integer rounding, which violates the stratified design's positive sample-size requirement.

Facts & Assumptions

Given: The fixed strata above, independent within-stratum SRSWOR, total sample size six, and componentwise rounding to the nearest integer.

[F1]

Stratified sampling designs and the stratified total estimator: A stratified design requires integer sample sizes $1 \leq n_{h} \leq N_{h}$ ; for a noncertainty stratum, $S_{h}^{2} = (N_{h} - 1)^{- 1} \sum_{i \in U_{h}} (y_{i} - {\overset{ˉ}{y}}_{h})^{2}$ .

[F2]

Continuous fixed-size Neyman allocation minimizes stratified variance: With positive noncertainty stratum variances and a fixed positive total sample size, the uncapped continuous optimum is proportional to $N_{h} S_{h}$ ; this relaxation imposes no integer, capacity, lower-bound, or minimum-sample constraints.

[F3]

Unbiasedness and variance of the stratified total estimator: For an admissible allocation, the stratified total's variance is the sum of the stratum terms $N_{h}^{2} (1 - n_{h} / N_{h}) S_{h}^{2} / n_{h}$ over noncertainty strata.

Verification

1.1givenchoosealgebra

The first stratum has mean zero and $S_{1}^{2} = ((- 1 / \sqrt{2})^{2} + (1 / \sqrt{2})^{2}) / (2 - 1) = 1$ . The second has mean zero and $S_{2}^{2} = 10 (9 / \sqrt{10})^{2} / (10 - 1) = 9$ , so $S_{2} = 3$ .

2.1F1F2step 1.1algebra

The Neyman weights are $N_{1} S_{1} = 2$ and $N_{2} S_{2} = 30$ . By [F2] and step 1.1, $x_{1} = 6 (2) / (2 + 30) = 3 / 8$ and $x_{2} = 6 (30) / (2 + 30) = 45 / 8$ . Nearest-integer rounding gives $(0, 6)$ , which violates [F1]'s $n_{1} \geq 1$ condition.

3.1F1F3step 1.1step 2.1algebra∎

Allocation $(n_{1}, n_{2}) = (1, 5)$ is admissible. By [F3] and step 1.1 its variance is $2^{2} (1 - 1 / 2) 1 / 1 + 10^{2} (1 - 5 / 10) 9 / 5 = 2 + 90 = 92$ ; this feasible correction is not claimed to solve every constrained integer problem.

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Sources

Shalabh, Sampling Theory, MTH 417 lecture notes, Chapter 4: Stratified Sampling, PDF pp. 8–9 and 16 (standard reference, not scraped)
Robin Arnold, STAT392 Sample Surveys, Chapter 8: Stratified Sampling, §8.8 and Notes 1–2 (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, §§2.1–2.3 (standard reference, not scraped)