LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

Each fixed-point KDE evaluation is a statistic

Statement

Fix $x \in R^{d}$ , $h > 0$ , $n \geq 1$ , and a Borel normalized kernel $K$ . The evaluation map $T_{x} : (R^{d})^{n} \to R, (z_{1}, \dots, z_{n}) \mapsto \frac{1}{n h^{d}} \sum_{i = 1}^{n} K ⁣ (\frac{x - z_{i}}{h})$ is measurable for the finite product Borel sigma-algebra. Hence it is a statistic on the iid density experiment.

Facts & Assumptions

Given: Fixed $x \in R^{d}$ , $h > 0$ , $n \geq 1$ , and a Borel normalized kernel $K$ .

[F1]

The sample space is the finite product Borel space and $K$ is Borel (Density targets and the kernel density estimator).

[F2]

A statistic is a measurable map from the experiment's data space to its measurable target (Statistics and estimators, A measurable function between measurable spaces).

[F3]

Compositions of measurable maps are measurable (Composition with a Borel measurable outer map preserves measurability).

[F4]

Finite sums and scalar multiples of real measurable functions are measurable (Arithmetic and lattice operations preserve measurability whenever they are defined).

Proof

1.1givenF1F3

Each coordinate projection $π_{i} : (R^{d})^{n} \to R^{d}$ is measurable by the definition of the finite product sigma-algebra. The map $z \mapsto (x - z) / h$ is continuous, hence Borel; composing it with $π_{i}$ and then with $K$ gives a real measurable function on the sample space by [F1] and [F3].

2.1step 1.1F2F4∎

By [F4], the finite sum of these functions and its scalar multiple $(n h^{d})^{- 1}$ are measurable. This is exactly $T_{x}$ , so [F2] makes it a statistic with target $R$ . Since the estimand $ψ_{x}$ also has target space $R$ , $T_{x}$ is an estimator of $ψ_{x}$ .

Depends on

Used by

Finite-bandwidth Gaussian smoothing Example
Gaussian KDE boundary bias for a uniform density Example
Pointwise second-order bias of the KDE Theorem
Pointwise variance of the KDE Theorem

Cited to discharge well-definedness by Density targets and the kernel density estimator.

Dependency tree · two levels

24 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

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OpenCourseWare 14.385, Fall 2007, PDF pp. 1–3 (standard reference, not scraped)
Adrian Bowman and Ludger Evers, Nonparametric Smoothing, APTS 2013–14 course notes, §1.2, PDF pp. 6–7 (standard reference, not scraped)