ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 non-injective binding map cannot recover a sign

Example

Let $Θ = {- 1, 1}$ , $X = {- 1, 1}$ with its full sigma-algebra, and let $P_{θ}$ be the point mass at $Y = θ$ . Take the statistic $S (y) = y^{2}$ .

Facts & Assumptions

Given: The finite experiment, statistic and moment-binding target above.

[F1]

The binding function is $b (θ) = E_{θ} [S]$ , and the target minimizes $∥ b (θ) - s ∥_{2}^{2}$ with the declared lexicographic tie-break. Binding function and indirect-inference target

[F2]

The compact-domain minimum and its lexicographic selector exist under the definition’s stated continuity hypotheses. The compact binding target exists and has a unique lexicographic tie-break

[F3]

The parameter identity is identifiable exactly when distinct parameter labels give distinct sampling laws. Identifiability of an estimand

[F4]

In a parametric model, identifiability is equivalent to constancy on observational-equivalence classes; for the identity target this is injectivity of the model map. Identifiability is constancy on observational classes

[F5]

The summary law assigns to each Borel set $B$ the probability $P_{θ} (S^{- 1} (B))$ . Law or distribution of a random element

Verification

1.1F1givenalgebra

Under either point-mass law, $S (Y) = Y^{2} = 1$ surely. Thus $b (- 1) = b (1) = 1$ .

1.2F3F4givenalgebra

The original laws differ: $P_{- 1} ({- 1}) = 1$ but $P_{1} ({- 1}) = 0$ . Therefore the parameter identity is identifiable in the original experiment.

2.1F1F2step 1.1algebra

For the observed summary $s = 1$ , $q_{1}$ is the constant continuous function zero on the finite compact set $Θ$ , so $M_{1} = {- 1, 1}$ . The target’s lexicographic rule selects $- 1$ .

3.1F4F5step 1.1algebra∎

By the pushforward-law definition [F5], both summary laws assign probability one to ${1}$ , so the summary-law map is not injective and the parameter identity is not identifiable from this summary. The moment target therefore does not recover the sign, even though the raw experiment distinguishes it.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Christian Gouriéroux, Alain Monfort, and Eric Renault, Indirect Inference, Journal of Applied Econometrics 8, Supplement, S85–S118 (1993) (standard reference, not scraped)
Thomas J. Sargent and John Stachurski, Intermediate Quantitative Economics with Python, Chapter 38: Which Moments to Match? (standard reference, not scraped)