= Box-kernel bias of a piecewise constant density
{title2=$\|K_h*f-f\|_2^2=\frac h{12}\sum_r\Delta_r^2$}
For a density with finitely many jumps $\Delta_r$ at separated points and the <unit-width box kernel>, take $h$ smaller than all consecutive jump spacings. The smoothing bias is supported in disjoint length-$h$ neighbourhoods of the jumps. At each jump its two sides form triangles of height $|\Delta_r|/2$, and exact integration gives
$$
\|K_h*f-f\|_2^2=\frac h{12}\sum_r\Delta_r^2.
$$
Combined with the <integrated variance of a kernel density estimator>, this gives expected $L^2$ error $O(n^{-1/4})$ at $h\asymp n^{-1/2}$.
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