Nadaraya–Watson bias from design gradients and boundaries

Given: The two models below and $K(u)=\frac{1}{2}{\mathbf{1}}_{\{\mathrm{\mid }u\mathrm{\mid }\le 1\}}$, which is symmetric, nonnegative, normalized, and has ${\mu }_2=1\mathrm{/}3$.

Under its stated interior hypotheses, Pointwise population bias of the Nadaraya–Watson smoother identifies the population smoother by the kernel-weighted ratio and gives the interior expansion used in step 1.1.

The regression target is a chosen version of $\mathbb{E}[Y\mid \sigma (X)]$. Regression target and chosen version.

A density $f$ defines the measure $fd\lambda$ by $(fd\lambda )(A)={\int }_{A}fd\lambda$. The measure with density $f$ relative to $\mu$.

In the interior model, $f(t)=(1+t)\mathrm{/}2$ on $[-1,1]$ and $Y=X$. Choose the regular conditional law $Q(t,dy)={\delta }_t(dy)$ for every $t\in \mathbb{R}$, giving the pointwise version $m(t)=t$; also $f^{\mathrm{\prime }}(0)\mathrm{/}f(0)=1$.

In the boundary model, $X$ is uniform on $[0,1]$, $Y=X$, and the prediction point is the support boundary $x=0$. Choose the regular conditional law $Q(t,dy)={\delta }_t(dy)$ for every $t\in \mathbb{R}$, giving the pointwise version $m(t)=t$.

AC is inherited through the chosen versions of the regression target The Axiom of Choice; the finite integrations use no choice.