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

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Local-linear estimator can lack a finite-sample expectation

Statement refuted

At a fixed one-dimensional interior point, suppose the design density is positive and $C^{2}$ locally, the chosen regression function is $C^{2}$ locally, and the conditional response variance is finite and continuous. Suppose $K$ is nonnegative, symmetric, positive near zero, normalized, with finite positive second moment and finite squared integral. If $h_{n} \to 0$ , $n h_{n} \to \infty$ and $n h_{n}^{3} \to \infty$ , then the empirical local-linear fit with a fixed fallback only on singular designs has a Lebesgue expectation (and hence an unconditional second-order mean-bias expansion).

Facts & Assumptions

Given: $X_{i}$ are independent uniform variables on $[- 1, 1]$ ; $Y_{i}$ are independent Rademacher variables independent of all predictors; $K (u) = \frac{1}{2} 1_{{∣ u ∣ \leq 1}}$ ; $h_{n} = n^{- 1 / 4}$ ; and the degree-one estimator has any fixed fallback.

[F1]

A degree-one local-polynomial estimator is the intercept of the weighted least-squares coefficient vector when its moment matrix is nonsingular, and uses its fallback only when that matrix is singular. Local-polynomial estimator.

[F2]

The chosen regression version represents $E [Y ∣ σ (X)]$ almost surely. Regression target and chosen version.

[F3]

Independent and identically distributed variables have a common law and are mutually independent. Identical distribution and IID families.

[F4]

A density $f$ defines the measure $f d λ$ by $(f d λ) (A) = \int_{A} f d λ$ . The measure with density $f$ relative to $μ$ .

[A1]

AC is used only to specify the conditional-law version The Axiom of Choice; all sample-space and divergence calculations are choice-free.

[A2]

Choose the regular conditional law $Q (t, \cdot) = \frac{1}{2} δ_{- 1} + \frac{1}{2} δ_{1}$ for every $t \in R$ . This is a version of the response law given $X$ and gives the selected version $m (t) = 0$ and conditional variance $1$ at every $t$ .

Counterexample

1.1givenF1F2F3F4A1A2construct

Put $h = h_{n}$ . By [A2], the chosen regression function is $m (t) = 0$ and the conditional response variance is $1$ at every $t$ ; the uniform design density is $1 / 2$ near $0$ . The uniform kernel has $μ_{2} = 1 / 3$ and $R (K) = 1 / 2$ . For each $n \geq 2$ , the event $E_{n} = {X_{1} \in [h / 4, h / 3], X_{2} = X_{1} + δ for 0 < δ < h / 12, ∣ X_{j} ∣ > h for j \geq 3}$ has positive design density; on it exactly two kernel weights are positive. Since $X_{1} \neq X_{2}$ , the two-column degree-one design matrix is invertible, so [F1] gives its intercept $(X_{2} Y_{1} - X_{1} Y_{2}) / (X_{2} - X_{1})$ .

2.1givenstep 1.1F2algebra

Conditional on $E_{n}$ , each sign pattern $(Y_{1}, Y_{2}) = (1, - 1)$ and $(- 1, 1)$ has probability $1 / 4$ , and the intercepts are respectively $(X_{1} + X_{2}) / δ$ and $- (X_{1} + X_{2}) / δ$ . On the chosen $X_{1}$ interval, $X_{1} + X_{2} \geq h / 2$ , while the joint design density in $(X_{1}, δ)$ is a positive constant. Integrating either signed tail therefore includes a positive multiple of $\int_{0}^{h / 12} (h / 2) δ^{- 1} d δ = \infty$ . Both positive and negative parts have infinite expectation, so the estimator has no Lebesgue expectation.

3.1givenF1step 2.1algebra∎

The bandwidth satisfies $h_{n} \to 0$ , $n h_{n} = n^{3 / 4} \to \infty$ , and $n h_{n}^{3} = n^{1 / 4} \to \infty$ , and the matrix is nonsingular on $E_{n}$ . Thus the counterexample satisfies these local smoothness, kernel and bandwidth hypotheses while the finite-sample expectation does not exist; the singular-matrix fallback is not used on these samples.

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Sources

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