TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Continuous cost-optimal stratified allocation

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 variable noncertainty strata with $S_{h} > 0$ , and hold all other allocations fixed. Assign each $h \in J$ a positive per-unit cost $c_{h}$ . For a fixed positive residual cost budget $C^{⋆}$ , the uncapped continuous relaxation of the stratified variance expression, minimized over positive real allocations $x_{h} > 0$ for every $h \in J$ subject to $\sum_{h \in J} c_{h} x_{h} = C^{⋆}$ , has the unique minimizer $x_{h} = C^{⋆} \frac{N_{h} S_{h} / \sqrt{c_{h}}}{\sum_{k \in J} N_{k} S_{k} \sqrt{c_{k}}} (h \in J) .$ Thus $x_{h} \propto N_{h} S_{h} / \sqrt{c_{h}}$ . As in the fixed-size result, this continuous formula does not enforce integer, capacity, lower-bound, or minimum-sample constraints.

Facts & Assumptions

Given: A finite stratified SRSWOR design, a nonempty set $J$ of variable noncertainty strata with $S_{h} > 0$ , fixed allocations outside $J$ , per-unit costs $c_{h} > 0$ , and a positive residual cost budget $C^{⋆}$ .

[F1]

Stratified sampling designs and the stratified total estimator: For each noncertainty stratum, $S_{h}^{2}$ is the finite-population variance and $S_{h} = \sqrt{S_{h}^{2}}$ denotes its nonnegative standard deviation. The hypothesis $S_{h} > 0$ is imposed only for $h \in J$ .

[F2]

Unbiasedness and variance of the stratified total estimator: The variable variance term for stratum $h$ is $N_{h}^{2} (1 - n_{h} / N_{h}) S_{h}^{2} / n_{h}$ ; contributions from fixed strata do not depend on the variable allocation.

Proof

1.1givenF1F2algebra

Put $a_{h} = N_{h} S_{h} > 0$ . Replacing $n_{h}$ by $x_{h}$ in [F2] gives $N_{h}^{2} (1 - x_{h} / N_{h}) S_{h}^{2} / x_{h} = a_{h}^{2} / x_{h} - N_{h} S_{h}^{2}$ . The subtracted and fixed-stratum terms do not vary with $x$ , so it suffices to minimize $\sum_{h \in J} a_{h}^{2} / x_{h}$ subject to $\sum_{h \in J} c_{h} x_{h} = C^{⋆}$ .

2.1givenstep 1.1algebra

Set $p_{h} = a_{h} / \sqrt{x_{h}}$ and $q_{h} = \sqrt{c_{h} x_{h}}$ . 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 $\sum_{h} q_{h}^{2} = C^{⋆}$ and $\sum_{h} p_{h} q_{h} = \sum_{h} a_{h} \sqrt{c_{h}}$ , the objective is at least $(\sum_{h} a_{h} \sqrt{c_{h}})^{2} / C^{⋆}$ , with equality exactly when $a_{h} / (x_{h} \sqrt{c_{h}})$ is constant.

3.1givenstep 1.1step 2.1algebra∎

The allocation $x_{h}^{⋆} = C^{⋆} (a_{h} / \sqrt{c_{h}}) / (\sum_{k \in J} a_{k} \sqrt{c_{k}})$ is positive and satisfies $\sum_{h} c_{h} x_{h}^{⋆} = C^{⋆}$ . It makes $a_{h} / (x_{h}^{⋆} \sqrt{c_{h}})$ constant and attains step 2.1's bound; the equality condition uniquely determines it. By step 1.1 it uniquely minimizes the relaxed variance and gives the stated formula.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shalabh, Sampling Theory, MTH 417 lecture notes, Chapter 4: Stratified Sampling, PDF pp. 9–11 (standard reference, not scraped)
Robin Arnold, STAT392 Sample Surveys, Chapter 8: Stratified Sampling, §8.8 and Example 8.8.1 (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)