Mean absolute error of local constant regression ($\mathbb{E}|\hat{m}(x)-m(x)|\le 2\sigma /\sqrt{n\min (h,1)}+Lh/2$)

For an equally spaced design with independent errors of variance at most ${\sigma }^2$ and a regression function with Lipschitz constant $L$, the unit-width box kernel estimator averages the $N_x$ observations in the window. Its bias is at most $Lh/2$, and its variance is at most ${\sigma }^2/N_x$. The window occupancy for an equally spaced regression design gives, whenever $nh\ge 2$,In the usual range $h\le 1$, this is $O(\sigma /\sqrt{nh}+Lh)$ uniformly up to the boundary. For arbitrarily large bandwidth, the stochastic error cannot continue to decrease as $1/\sqrt{nh}$: once all observations are included the estimator is simply their mean.