= Second-order boundary expansion for local constant regression
{title2=$\operatorname{Bias}(\widehat m_h(\alpha h;0))=h\beta_1m'(\alpha h)+\tfrac12h^2\beta_2m''(\alpha h)+o(h^2)$}
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 $h\to0$, $nh^2\to\infty$, write $\mu_{r,\alpha}=\int_{-\alpha}^1u^rK(u)\,du$ and $\beta_r=\mu_{r,\alpha}/\mu_{0,\alpha}$. On the rescaled grid of spacing $1/(nh)$, the <Riemann sum> error for $u^rK(u)$ 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 $h\beta_1m'(0)+h^2(\alpha\beta_1+\beta_2/2)m''(0)+o(h^2)$. The first-order term explains the generic boundary bias of a <local constant estimator>.
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