ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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Gaussian KDE boundary bias for a uniform density

Example

In one dimension, choose the Borel density version $f = 1_{[0, 1]}$ and the standard Gaussian kernel $K (u) = (2 π)^{- 1 / 2} e^{- u^{2} / 2}$ . At the support boundary $x = 0$ , $E {\hat{f}}_{h} (0) = \int_{- 1 / h}^{0} K (u) d u ⟶ \frac{1}{2} (h ↓ 0),$ while the chosen target value is $f (0) = 1$ . Hence the uncorrected estimator has limiting bias $- 1 / 2$ at the boundary. The smoothness hypotheses of the pointwise second-order bias theorem fail at $x = 0$ .

Facts & Assumptions

Given: $d = 1$ , iid observations with chosen density version $f = 1_{[0, 1]}$ , kernel $K (u) = (2 π)^{- 1 / 2} e^{- u^{2} / 2}$ , bandwidth $h > 0$ , and evaluation point $x = 0$ .

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The estimator is ${\hat{f}}_{h} (x) = (n h)^{- 1} \sum_{i} K ((x - X_{i}) / h)$ (Density targets and the kernel density estimator).

[F2]

Expectation is integration against the sample law, which has density $f$ (Change of variables for expectation, Integrating against a density agrees with integrating the product).

[F3]

Expectation is linear on integrable variables (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F4]

The nonnegative plane Gaussian integral is $\int_{R^{2}} e^{- (x^{2} + y^{2})} d (x, y) = π$ (The plane Gaussian integral equals $π$ by polar coordinates).

[F5]

Tonelli factors nonnegative product integrals (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F6]

Nonnegative integrals pass through increasing limits (Monotone convergence for the integral).

[F7]

Under ${A C}_{ω}$ , affine C $^{1}$ substitutions obey nonnegative change of variables (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

[F8]

Under ${A C}_{ω}$ , a one-dimensional box has Lebesgue measure equal to its length (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

Verification

1.1givenA1F4F5F7F8algebra

The function $f = 1_{[0, 1]}$ is a Borel density version: [F8] gives $λ_{1} ([0, 1]) = 1$ . Put $I = \int_{R} e^{- t^{2}} d t$ . Tonelli [F5] and [F4] give $I^{2} = π$ , so $I = \sqrt{π}$ . Scaling with [F7] shows $\int_{R} (2 π)^{- 1 / 2} e^{- t^{2} / 2} d t = 1$ . Since this kernel is even, $\int_{- \infty}^{0} K (u) d u = 1 / 2$ .

2.1givenA1F1F2F3F7step 1.1

By [F1]–[F3] and the density formula, each summand is integrable because $K$ is bounded and the sample law is a probability measure. Thus $E {\hat{f}}_{h} (0) = h^{- 1} \int_{0}^{1} K (- t / h) d t .$ Apply [F7] on $R$ to $T (u) = - h u$ and $g (t) = 1_{[0, 1]} (t) K (- t / h)$ . It gives $\int_{0}^{1} K (- t / h) d t = h \int_{- 1 / h}^{0} K (u) d u,$ so the displayed expectation equals $\int_{- 1 / h}^{0} K (u) d u$ .

3.1step 1.1F6step 2.1

Along any sequence $h_{m} ↓ 0$ , the intervals $[- 1 / h_{m}, 0]$ increase to $(- \infty, 0]$ . Monotone convergence [F6] and the half-mass calculation in step 1.1 give $\int_{- 1 / h_{m}}^{0} K (u) d u ⟶ \int_{- \infty}^{0} K (u) d u = \frac{1}{2} .$

4.1givenstep 3.1algebra∎

The density version was chosen with $f (0) = 1$ , so step 3.1 gives $E {\hat{f}}_{h} (0) - f (0) \to - 1 / 2$ . The selected density is discontinuous at $0$ , so this calculation does not satisfy the smooth interior-point theorem's hypotheses.

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Sources

Whitney K. Newey, Nonparametric and Semiparametric Estimation, MIT OpenCourseWare 14.385, Fall 2007, PDF pp. 1–3 (standard reference, not scraped)
Ryan Tibshirani, Nonparametric Regression: Nearest Neighbors and Kernels, Advanced Topics in Statistical Learning, Spring 2024, §3.1, PDF pp. 5–7 (standard reference, not scraped)