Box-kernel bias of a piecewise constant density ($\Vert K_h\ast f-f{\Vert }_2^2=\frac{h}{12}{\sum }_r{\Delta }_r^2$)

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 givesCombined with the integrated variance of a kernel density estimator, this gives expected $L^2$ error $O(n^{-1/4})$ at $h\asymp n^{-1/2}$.