TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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A zero inclusion probability prevents total identification and universal unbiased estimation

Statement

Let $i \in U$ have $π_{i} = 0$ in a finite-population design as in Finite populations and sampling designs. Regard the total as the estimand $y \mapsto T (y)$ in the induced experiment. Then this estimand is not identifiable. Moreover, no estimator that is a function only of the observed record can be design-unbiased for every $y \in R^{U}$ . The estimator target is $R$ with its Borel sigma-algebra.

Facts & Assumptions

Given: A finite nonempty $U$ , a design support $S_{d}$ with $p_{d} (s) > 0$ for every $s \in S_{d}$ , a fixed $i \in U$ with $π_{i} = 0$ , and the induced observed-record experiment.

[F1]

Finite populations and sampling designs: The design support has strictly positive weights, $π_{i} = \sum_{s : i \in s} p_{d} (s)$ , the record is $O_{y} (s) = (s, y ∣_{s})$ , the common observed-data law is $P_{y}^{d}$ , and the total is $T (y) = \sum_{j \in U} y_{j}$ . The induced-data expectation is $E_{y}^{d} [h] = \sum_{s} p_{d} (s) h (O_{y} (s))$ .

[F2]

A finite-population design induces probability laws on observed records: Every $P_{y}^{d}$ is a probability measure on the common observed-record space.

[F3]

Parametric statistical models: The induced model is the map $P_{d} : y \mapsto P_{y}^{d}$ on parameter set $R^{U}$ .

[F4]

Estimands on a statistical model: An estimand is a function from the model's parameter set to its target set; here it is $ψ_{T} (y) = T (y)$ .

[F5]

Identifiability of an estimand: An estimand is identifiable only if equal sampling laws imply equal target values.

[F6]

Statistics and estimators: An estimator is a measurable statistic with the target space of the specified estimand.

[F7]

The Borel sigma-algebra of a topological space: $R$ has its Borel sigma-algebra, generated by the open subsets.

[F8]

Expectation of a real random variable on a finite probability space: Design expectation is the finite weighted sum over $S_{d}$ .

[F9]

Unbiased estimators and UMVU estimators: An estimator is unbiased if it is integrable under every model law and its expectation equals the estimand at every parameter.

Proof

1.1givenF1algebra

Since $π_{i} = \sum_{s : i \in s} p_{d} (s) = 0$ and every support weight is strictly positive by [F1], no $s \in S_{d}$ contains $i$ .

2.1F1F2F3F4F5step 1.1algebra

Fix any $y \in R^{U}$ and any $c \in R ∖ {0}$ . Define $y_{i}^{'} = y_{i} + c$ and $y_{j}^{'} = y_{j}$ for $j \neq i$ . By step 1.1, $O_{y} (s) = O_{y^{'}} (s)$ for every $s \in S_{d}$ . Therefore the finite sums defining $P_{y}^{d} (B)$ and $P_{y^{'}}^{d} (B)$ agree for every measurable record set $B$ , so $P_{y}^{d} = P_{y^{'}}^{d}$ ; [F2] ensures these are laws in the induced experiment. But $T (y^{'}) = T (y) + c \neq T (y)$ , so the total fails the defining implication for identifiability.

3.1F1F6F7F8F9step 2.1algebra∎

Let $δ$ be any estimator from the common record space to $(R, B (R))$ . Since $O_{y} (s) = O_{y^{'}} (s)$ for every support sample, [F8] gives $E_{y}^{d} [δ] = \sum_{s \in S_{d}} p_{d} (s) δ (O_{y} (s)) = \sum_{s \in S_{d}} p_{d} (s) δ (O_{y^{'}} (s)) = E_{y^{'}}^{d} [δ]$ . Both expectations are finite because the design support is finite and $δ$ is real-valued. By [F9], an estimator unbiased for every finite population would make these equal expectations equal both $T (y)$ and $T (y^{'})$ , which are unequal. Thus no such universally unbiased estimator exists.

Depends on

Used by

An unobserved unit prevents universal unbiased estimation Example

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Sources

Shalabh, Sampling Theory, Chapter 1: Introduction, IIT Kanpur (standard reference, not scraped)
John J. Borkowski, Horvitz–Thompson Estimators, STAT 446 Lecture 6, §6.2, Montana State University (standard reference, not scraped)