Second-order boundary expansion for local constant regression ($\mathrm{Bias}({\hat{m}}_h(\alpha h;0))=h{\beta }_1{m}^{\prime }(\alpha h)+\frac{1}{2}{h}^2{\beta }_2{m}^{\prime \prime }(\alpha h)+o(h^2)$)

For fixed $\alpha \in [0,1)$, design $x_i=i/n$, $m\in C^2[0,1]$, a nonzero nonnegative symmetric piecewise $C^1$ regression kernel supported on $[-1,1]$, and $h\to 0$, $n{h}^2\to \infty$, write ${\mu }_{r,\alpha }={\int }_{-\alpha }^1{u}^{r}K(u)du$ and ${\beta }_r={\mu }_{r,\alpha }/{\mu }_{0,\alpha }$. On the rescaled grid of spacing $1/(nh)$, the Riemann sum error for $u^{r}K(u)$ is $O((nh{)}^{-1})$, by bounded variation. The normalized first and second local moments are therefore $h{\beta }_1+O(n^{-1})$ and $h^2{\beta }_2+O(h/n)$. A uniform second-order Taylor expansion gives the displayed bias of an estimator, since $n^{-1}=o(h^2)$. Equivalently it is $h{\beta }_1{m}^{\prime }(0)+h^2(\alpha {\beta }_1+{\beta }_2/2)m^{\prime \prime }(0)+o(h^2)$. The first-order term explains the generic boundary bias of a local constant estimator.