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Pointwise population bias of the Nadaraya–Watson smoother

Statement

Assume AC for the chosen regression version Regression target and chosen version. Let $X$ have Lebesgue 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 that $K$ has compact support, or that $f, m$ and their first two derivatives are bounded and continuous on $R$ . In these first two alternatives, no continuity or boundedness of $K$ is required. The global alternative supplies tail control for noncompact kernels; local smoothness alone does not supply it. 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. For sufficiently small $h > 0$ , the population smoother corresponding to Nadaraya–Watson estimator is well defined by $m_{h} (x) = \frac{\int K (u) m (x - h u) f (x - h u) d u}{\int K (u) f (x - h u) d u},$ and $m_{h} (x) - m (x) = \frac{h^{2} μ_{2}}{2} (m^{''} (x) + 2 m^{'} (x) \frac{f^{'} (x)}{f (x)}) + o (h^{2}) .$

Facts & Assumptions

Given: The selected regression version, density, kernel, and either localization or global domination hypotheses in the statement. Integrals are Lebesgue integrals.

[F1]

The chosen Borel regression version represents the conditional mean almost surely. Regression target and chosen version.

[F2]

The Nadaraya–Watson estimator normalizes its nonnegative weights, with a fixed zero-denominator fallback. Nadaraya–Watson estimator.

[F3]

A density $f$ defines the 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

1.1givenF1F3F4algebra

Put $g = m f$ . For $v = f, g$ , twice applying the fundamental theorem of calculus gives $v (x - t) = v (x) - t v^{'} (x) + t^{2} \int_{0}^{1} (1 - s) v^{''} (x - s t) d s$ . Thus $v (x - h u) = v (x) - h u v^{'} (x) + \frac{1}{2} h^{2} u^{2} v^{''} (x) + h^{2} u^{2} r_{v, h} (u)$ . On compact kernel support the remainder tends uniformly to zero. Under the global alternative it is uniformly bounded and tends to zero for each fixed $u$ : split the remainder integral at $∣ u ∣ \leq A$ , use uniform continuity near $x$ on that fixed set, and bound the remaining part by $C \int_{∣ u ∣ > A} u^{2} K (u) d u \to 0$ as $A \to \infty$ . In the rapid-tail alternative [F4] gives $\int ∣ g ∣ < \infty$ . Put $ϵ_{h} = \sup_{∣ u ∣ \geq δ / h} ∣ u ∣^{3} K (u) \to 0$ . For $v = f, m f$ , $\int ∣ v ∣ < \infty$ and $h^{- 1} \int_{∣ z - x ∣ > δ} K ((x - z) / h) ∣ v (z) ∣ d z \leq ϵ_{h} h^{2} δ^{- 3} \int ∣ v ∣ = o (h^{2})$ . On $∣ z - x ∣ \leq δ$ use the local Taylor remainder; its normalized integral is bounded by $μ_{2}$ times the modulus of continuity of $v^{''}$ at $x$ . The omitted constant, linear and quadratic Taylor tails are $o (h^{2})$ because $\int_{∣ u ∣ > δ / h} u^{2} K \to 0$ . First let $h \to 0$ , then $δ \to 0$ . In every alternative the integrated remainder is $o (h^{2})$ . Symmetry and the finite second moment give $\int u K (u) d u = 0$ , so $\int K (u) v (x - h u) d u = v (x) + \frac{1}{2} h^{2} μ_{2} v^{''} (x) + o (h^{2})$ . All integrals are finite by localization or the global bounds.

2.1givenF2step 1.1algebra∎

Apply step 1.1 to numerator $g$ and denominator $f$ . The latter tends to $f (x) > 0$ . Taking their quotient and subtracting $m (x)$ gives $\frac{1}{2} h^{2} μ_{2} {(m f)^{''} / f - m f^{''} / f} (x) + o (h^{2}) = \frac{1}{2} h^{2} μ_{2} (m^{''} + 2 m^{'} f^{'} / f) (x) + o (h^{2})$ , as asserted.

Depends on

Used by

Nadaraya–Watson bias from design gradients and boundaries Counterexample
Finite-sample mean bias of the Nadaraya–Watson estimator Theorem

Dependency tree · two levels

12 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 course notes (standard reference, not scraped)
Ryan Tibshirani, Nonparametric Regression: Nearest Neighbors and Kernels (standard reference, not scraped)