ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A same-sample fitted coefficient can bias the estimator

Example

Let $X_{i}$ be IID uniform on ${- 1, 0, 1}$ , let $f (x) = x^{2}$ , $h (x) = x$ , and use $θ = E h (X) = 0$ . For $n = 2$ , fit the scalar coefficient on the same two observations by $\hat{β} = \overset{}{Cov} (h, f) / \overset{}{Var} (h)$ when the sample variance is positive, and set it to $0$ when the sample variance is zero. The resulting control estimator is biased.

Facts & Assumptions

Given: The three-point uniform law, sample size two, the specified $f, h, θ$ , and the zero-denominator convention.

[F1]

Same-sample coefficient remainder is of order $O_{P} (n^{- 1})$ : The same-sample coefficient is the inverse sample control covariance times the sample cross-covariance when defined, with a zero fallback when singular.

[F2]

Same-sample coefficient remainder is of order $O_{P} (n^{- 1})$ defines the plug-in estimator pointwise; for a scalar fitted coefficient it is $\overset{ˉ}{f} - \hat{β} (\overset{ˉ}{h} - θ)$ .

Verification

Given: Each of the nine ordered pairs $(X_{1}, X_{2}) \in {- 1, 0, 1}^{2}$ has probability $1 / 9$ .

1.1F1F2givenalgebra

On the three equal pairs, the fallback coefficient is zero and the estimates are $1, 0, 1$ . On the two ordered pairs $(- 1, 1)$ and $(1, - 1)$ , the fitted slope is zero and each estimate is $1$ . On each of the four ordered adjacent pairs $(- 1, 0), (0, - 1), (0, 1), (1, 0)$ , the fitted slope is respectively $- 1, - 1, 1, 1$ , and direct substitution into [F2] gives estimate $0$ .

2.1givenstep 1.1algebra∎

Exactly four of the nine equally likely pairs yield estimate $1$ and the other five yield $0$ , so the expected estimate is $4 / 9$ . The target is $E f (X) = \frac{1}{3} (1 + 0 + 1) = 2 / 3$ , hence this same-sample fitted-coefficient estimator is biased.

Depends on

Used by

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