Interior absolute-error rate of the Nadaraya-Watson estimator ($\mathbb{E}|{\hat{m}}_h(x)-m(x)|=O(h^2+(nh{)}^{-1/2})$)

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 $G_n=(nh{)}^{-1}{\sum }_i(Y_i-m(x))1_{\{|X_i-x|\le h/2\}}$. Symmetry cancels its linear Taylor term, giving $\mathbb{E}{G}_n=O(h^2)$; independence gives $\mathrm{Var}{G}_n=O((nh{)}^{-1})$. 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.