Boundary bias of local constant regression ($\mathrm{Bias}(\hat{m}(0))\asymp h$)

A local constant average reproduces constants but need not reproduce a linear regression function near an endpoint. For $x_i=i/n$, $K=1_{[-1,1]}$, $m(x)=x$, and $nh\to \infty$, its deterministic estimate at zero is $(\lfloor nh\rfloor +1)/(2n)\sim h/2$. A local linear fit removes this first-order boundary bias of an estimator.