Mean absolute error bound for local polynomial regression ($\mathbb{E}|\hat{m}(x)-m(x)|\le C((nh{)}^{-1/2}+h^s)$)

For equally spaced fixed-design nonparametric regression, independent mean-zero uniformly finite-variance errors, a bounded compactly supported kernel, and a uniformly invertible local Gram matrix, degree $\ell \ge s-1$ gives the displayed bound for $s$ bounded derivatives. Kernel weights have absolute sum $O(1)$ and squared sum $O((nh{)}^{-1})$. Polynomial reproduction removes a degree-$s-1$ Taylor polynomial. Without the degree requirement the general bias of an estimator is only $O(h^{\min (s,\ell +1)})$.