Integrated squared bias from a density jump ($\int {\mathrm{Bias}}^2\asymp h$)

An uncorrected nonnegative symmetric kernel density estimator can leak an order-one amount across a jump in a probability density function. If its transition region has width proportional to the smoothing bandwidth $h$, the integrated mean squared error includes squared bias of an estimator of order $h$, rather than the order $h^4$ familiar for twice-smooth interior densities. For the unit-width uniform window and the exponential distribution of rate one, the integrated squared bias of an estimator is $h/12+o(h)$.

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 $h\asymp n^{-1/2}$.