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Pointwise population bias of local-linear regression

Statement

Assume AC for the chosen regression version Regression target and chosen version. Let $X$ have density $f$ , with $f (x) > 0$ at a fixed interior point $x$ , and let $f, m$ be $C^{2}$ near $x$ . Let $K$ be a finite-valued Borel, nonnegative symmetric kernel with $\int K = 1$ , $0 < μ_{2} = \int u^{2} K (u) d u < \infty$ and $\int K^{2} < \infty$ . Assume either compact kernel support, or that $f, m$ and their first two derivatives are bounded and continuous on $R$ . A further localization alternative, including the Gaussian kernel, is that $E [Y^{2}] < \infty$ , $K$ is bounded, and $\lim_{A \to \infty} \sup_{∣ u ∣ \geq A} ∣ u ∣^{3} K (u) = 0.$ Under this rapid-tail alternative only the stated local smoothness of $f, m$ is needed; no global boundedness of $m$ or of its derivatives is assumed. Set $S_{j} (h) = \int K (u) u^{j} f (x - h u) d u, T_{j} (h) = \int K (u) u^{j} f (x - h u) m (x - h u) d u .$ For sufficiently small $h$ , $D_{h} = S_{0} S_{2} - S_{1}^{2} > 0$ and the population local-linear intercept, in the coordinates of Local-polynomial estimator, is $m_{L L, h} (x) = \frac{S_{2} T_{0} - S_{1} T_{1}}{D_{h}} .$ Then $m_{L L, h} (x) - m (x) = \frac{h^{2} μ_{2}}{2} m^{''} (x) + o (h^{2}) .$ This requires no fourth kernel moment.

Facts & Assumptions

Given: The density, selected version, kernel moments and localization or global domination hypotheses in the statement.

[F1]

The local-polynomial fit returns the intercept of the weighted normal equations on a nonsingular design. Local-polynomial estimator.

[F2]

The selected regression function is a Borel conditional-mean version. Regression target and chosen version.

[F3]

A density supplies its integration law $f d λ$ . The measure with density $f$ relative to $μ$ .

[F4]

For $P_{X}$ -almost every $t$ , $m (t) = \int y Q (t, d y)$ , and conditional integration gives $\int ∣ m (t) ∣ f (t) d t \leq E ∣ Y ∣ < \infty$ by the triangle inequality. Conditional expectation as a measurable function of the conditioning variable

Proof

1.1givenF2F3F4algebra

Write $r_{h} (u) = m (x - h u) - m (x) + h u m^{'} (x)$ . Taylor's integral formula bounds $∣ r_{h} (u) ∣ \leq C h^{2} u^{2}$ and gives $r_{h} (u) / h^{2} \to \frac{1}{2} m^{''} (x) u^{2}$ pointwise. It also gives $∣ r_{h} (u) ∣ \leq C h ∣ u ∣$ and $r_{h} (u) / (h ∣ u ∣) \to 0$ for each nonzero $u$ . These bounds hold on compact kernel support for small $h$ , or globally under the alternative assumptions. With $R_{j} = \int K (u) u^{j} f (x - h u) r_{h} (u) d u$ , splitting at a fixed $∣ u ∣ \leq A$ and controlling the tails by $\int_{∣ u ∣ > A} u^{2} K (u) d u$ proves $R_{0} = \frac{1}{2} h^{2} m^{''} (x) f (x) μ_{2} + o (h^{2})$ and $R_{1} = o (h)$ . The same split yields $S_{0} \to f (x)$ , $S_{2} \to f (x) μ_{2}$ ; symmetry and the first-order expansion of $f$ give $S_{1} = - h f^{'} (x) μ_{2} + o (h) = O (h)$ . In the rapid-tail alternative [F4] gives $\int ∣ m ∣ f < \infty$ . Write $z = x - h u$ and split at $∣ z - x ∣ = δ$ . The far parts of $S_{0}, S_{1}, S_{2}$ are respectively $o (h^{2}), o (h), o (1)$ , since $K ((x - z) / h) \leq ϵ_{h} h^{3} / ∣ x - z ∣^{3}$ and $\int f = 1$ . The far parts of $R_{0}, R_{1}$ are $o (h^{2}), o (h)$ : expand $r = m (z) - m (x) - m^{'} (x) (z - x)$ and use $\int ∣ m ∣ f < \infty$ with $∣ z - x ∣ \geq δ$ . On the near region the preceding local bounds and the integrable $u^{2} K$ apply; letting $δ \to 0$ after $h \to 0$ gives the same remainder limits. These estimates also make every stated integral finite.

2.1givenF1step 1.1algebra∎

The determinant tends to $f (x)^{2} μ_{2} > 0$ . Since $T_{j} = m (x) S_{j} - h m^{'} (x) S_{j + 1} + R_{j}$ for $j = 0, 1$ , the affine terms in the intercept cancel exactly and $m_{L L, h} (x) - m (x) = (S_{2} R_{0} - S_{1} R_{1}) / D_{h}$ . Step 1.1 makes $S_{1} R_{1} = o (h^{2})$ and gives the asserted coefficient after division by the positive limiting determinant.

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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)