Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 29J / c / Solution

Let $h=n^{-1/3}$. Away from the boundary, the estimator isIts variance is at mostand the bias of a kernel density estimator is bounded by $h\Vert f^{\prime }{\Vert }_{\infty }/2$, so its squared bias is at most $h^2/4$. This gives the claimed bound, with room to spare, whenever the full window lies in the support.

At $x=0$, however, the assertion is false as printed. For examplesatisfies $\sup (|f|+|f^{\prime }|)=3/4<1$, butwhereas $f(0)=1/2$. The mean-square error therefore tends to $1/16$, contradicting a bound tending to zero. The intended statement needs $x\ge h/2$, a boundary correction, or a condition such as $f(0)=0$.