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.

Regression target and chosen version

Definition

Assume the Axiom of Choice. Let $X, Y$ be real random variables on a probability space, with $E ∣ Y ∣ < \infty$ , and choose a regular conditional law $Q (x, d y)$ of $Y$ given $X = x$ . On $D = {x : \int ∣ y ∣ Q (x, d y) < \infty}$ define $m (x) = \int y Q (x, d y)$ , and put $m (x) = 0$ off $D$ . By Conditional expectation as a measurable function of the conditioning variable, $m$ is Borel measurable and $m (X)$ is a version of $E [Y ∣ σ (X)]$ . A regression target is the resulting $P_{X}$ -almost-sure equivalence class; a chosen regression-function version is the particular Borel function selected by the specified conditional law.

Pointwise smoothness is an additional hypothesis on the chosen version. It is not implied by existence of conditional expectation, and statements about $m (x)$ at a specified point use that selected version.

Source notes

Newey, “Nonparametric Regression,” PDF p. 8; Tibshirani, §1.1, PDF p. 1.

Depends on

Used by

Sobolev rate for spline-series training risk Corollary
Local-linear estimator can lack a finite-sample expectation Counterexample
Nadaraya–Watson bias from design gradients and boundaries Counterexample
Local-polynomial estimator Definition
Nadaraya–Watson estimator Definition
Pointwise squared-error risk for regression Definition
Local-linear regression reproduces affine responses Example
Conditional asymptotic variance of Nadaraya–Watson and local-linear estimators Theorem
Conditional-on-design bias of the empirical local-linear estimator Theorem
Finite-sample mean bias of the Nadaraya–Watson estimator Theorem
Pointwise population bias of local-linear regression Theorem
Pointwise population bias of the Nadaraya–Watson smoother Theorem
Pointwise risk is variance plus squared bias Theorem

Dependency tree · two levels

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Sources

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT course notes (standard reference, not scraped)
Ryan Tibshirani, Nonparametric Regression: Nearest Neighbors and Kernels (standard reference, not scraped)