DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Inclusion-covariance difference

Definition

Let $(U, y)$ be a finite population with a sampling design as in Finite populations and sampling designs. For each ordered pair $(i, j) \in U \times U$ , the inclusion-covariance difference is $Δ_{i j} : = π_{i j} - π_{i} π_{j},$ where $π_{i}$ is the probability that unit $i$ is included and $π_{i j}$ is the probability that both $i$ and $j$ are included. The supplier convention $π_{i i} = π_{i}$ gives $Δ_{i i} = π_{i} (1 - π_{i})$ .

Depends on

Finite populations and sampling designs

Used by

Positive pair covariances and a negative SYG estimate Example
Horvitz–Thompson variance and pair-inclusion estimate Theorem
Sen–Yates–Grundy variance identity Theorem
SRSWOR inclusion covariance and sample-mean variance Theorem

Dependency tree · two levels

10 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, Chapter 7: Varying Probability Sampling, PDF pp. 25–27 (standard reference, not scraped)
John J. Borkowski, Horvitz–Thompson Estimators, STAT 446 Lecture 6, §6.2, printed pp. 141–143 (standard reference, not scraped)