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.

Linearly estimable contrasts in fixed-design regression

Definition

In a fixed-design first-moment regression model with parameter $β \in R^{p}$ , a linear contrast is the estimand $ψ_{c} (β) : = c^{T} β, c \in R^{p} .$ The contrast is linearly estimable if there is a fixed vector $a \in R^{n}$ such that the statistic $a^{T} Y$ is unbiased for $ψ_{c}$ at every parameter value: $E_{β} [a^{T} Y] = c^{T} β for every β \in R^{p} .$ The vector $a$ may depend on the fixed design and on $c$ , but not on the observed response or on the unknown parameter. The map $y \mapsto a^{T} y$ is a measurable statistic by Linear statistics on Euclidean data are Borel-measurable and Statistics and estimators.

Depends on

Used by

Unequal error variances can make OLS fail to be BLUE Counterexample
Collinear predictors can leave a contrast estimable Example
Gauss–Markov theorem for estimable contrasts Theorem
Linear estimability is equivalent to row-space membership Theorem

Dependency tree · two levels

25 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

Statistics: Lecture Three — Linear Models (standard reference, not scraped)
Statistics for Applications, Section 15: Multiple Linear Regression (standard reference, not scraped)