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

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.

For the unit-width box kernel, the convolution and kernel density estimator areandIn particular $\mathbb{E}{\hat{f}}_{n,h}^K=g_h$. The scaling gives $\Vert K_h{\Vert }_2^2=h^{-1}\Vert K{\Vert }_2^2=h^{-1}$. Even if $f$ is not square-integrable, Young's convolution inequality gives $g_h\in L^2$ because $f\in L^1$ and $K_h\in L^2$.

Using independence to eliminate the cross terms in the centred estimator, and Tonelli theorem to integrate the nonnegative variance, gives the exact integrated variance of a kernel density estimator:The Cauchy-Schwarz inequality now yieldsThis constant comes from the unit $L^2$ norm of the unscaled unit-width box kernel.

For the piecewise constant probability density function, integrability forces the constants on both unbounded outer intervals to be zero. Thus $f$ is bounded, has compact support, and has finitely many jumps. Let ${\Delta }_r=f(x_r+)-f(x_r-)$ denote the jump at $x_r$. For $h$ smaller than the least gap between consecutive breakpoints, the smoothing regions $[x_r-h/2,x_r+h/2]$ do not overlap. Outside these regions the convolution equals $f$.

Within a region, put $v=x-x_r$. For $-h/2<v<0$, the bias is ${\Delta }_r(1/2+v/h)$; for $0<v<h/2$, it is $-{\Delta }_r(1/2-v/h)$. Endpoint values do not affect an $L^2$ norm. Integrating these two triangular errors givesThis is the box-kernel bias of a piecewise constant density. By the triangle inequality, with $C_f=({\sum }_r{\Delta }_r^2/12{)}^{1/2}$,Balancing this bias-variance tradeoff with $h_n=n^{-1/2}$ gives, for all sufficiently large $n$,

For the unit-width box kernel, the order-zero local polynomial estimator minimizes, over constants $a$,A common factor such as $h^{-1}$ in these weights does not change the minimizer. Put $I_x=\{i:|i/n-x|\le h/2\}$ and $N_x=|I_x|$. If $N_x>0$, differentiating the quadratic gives its unique minimizer:The last expression is the Nadaraya–Watson estimator. Every $x\in [0,1]$ is within $1/n$ of some design point $i/n$. Since $nh\ge 2$ gives $h/2\ge 1/n$, that point is in the closed kernel window. Thus $N_x\ge 1$ and the equality holds throughout the domain, including $x=0$ and $x=1$.

For the error bound, first take the usual bandwidth range $0<h\le 1$. The clipped window $[x-h/2,x+h/2]\cap [0,1]$ has length at least $h/2$. Any closed interval of length $L$ in $[0,1]$ contains at least $\lfloor nL\rfloor$ points of the design $\{1/n,\dots ,1\}$. Consequentlywhere the last inequality uses $nh/2\ge 1$. This is a window occupancy for an equally spaced regression design bound that remains valid at the boundary.

Let $L_m=\Vert m^{\prime }{\Vert }_{\infty }$, the Lipschitz constant supplied by the bounded-derivative assumption. The bias is bounded byIndependence and the variance bound giveBy the Cauchy-Schwarz inequality and the triangle inequality,Thus $\kappa =2$ works uniformly in $x$ and the design size in this bandwidth range. The same clipped-length proof in fact works for $0<h\le 2$.

There is a genuine omission in the unrestricted formulation: an upper restriction on bandwidth is necessary for the displayed $1/\sqrt{nh}$ variance scale. Take $n=1$, $m\equiv 0$, $Y_1\sim N(0,{\sigma }^2)$ and $h\ge 2$. The window contains the sole observation for every $x\in [0,1]$, and $nh\ge 2$ holds. Its mean absolute error is $\sigma \sqrt{2/\pi }$, whereas the proposed variance term is $\kappa \sigma /\sqrt{h}$. No fixed $\kappa$ can satisfy this for unbounded $h$.

A valid statement for every $h>0$ satisfying $nh\ge 2$ uses $q=\min (h,1)$. The clipped window has length at least $q/2$, so $N_x\ge nq/2-1$, while the preceding coverage argument gives $N_x\ge 1$. If $nq\ge 4$, the first bound gives $N_x\ge nq/4$; if $nq<4$, the second does. Therefore the same bias and variance calculation proves the unrestricted correctionThis mean absolute error of local constant regression bound reduces to the requested rate for ordinary small bandwidths and saturates its stochastic term when the window covers the whole design.