= Interior absolute-error rate of the Nadaraya-Watson estimator
{title2=$\mathbb E|\widehat m_h(x)-m(x)|=O(h^2+(nh)^{-1/2})$}
For the unit-width symmetric box <kernel for density estimation>, suppose the design density is positive and differentiable at $x$, the regression function is twice differentiable there, and the conditional variance and design density are bounded near $x$. Write $G_n=(nh)^{-1}\sum_i(Y_i-m(x))\mathbf1_{\{|X_i-x|\le h/2\}}$. Symmetry cancels its linear Taylor term, giving $\mathbb EG_n=O(h^2)$; <independence> gives $\operatorname{Var}G_n=O((nh)^{-1})$. The <absolute error bound for a ratio estimator with a controlled denominator> gives the displayed rate if the small-denominator contribution is negligible. The balance $h\asymp n^{-1/5}$ gives $O(n^{-2/5})$. Interior design positivity matters: support boundaries can prevent the linear cancellation.
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