= Mean absolute error of local constant regression
{title2=$\mathbb E|\widehat m(x)-m(x)|\le2\sigma/\sqrt{n\min(h,1)}+Lh/2$}
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 $nh\ge2$,
$$
\mathbb E|\widehat m(x)-m(x)|\le\frac{2\sigma}{\sqrt{n\min(h,1)}}+\frac{Lh}{2}.
$$
In the usual range $h\le1$, 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.
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