 Solution

Let $r=\lceil \beta \rceil -1$. The Hölder class $\mathcal{H}(\beta ,L)$ consists of functions with $r$ continuous derivatives whose $r$th derivative satisfieswith the equivalent Lipschitz convention when $\beta$ is an integer.

The degree-$p$ local polynomial regression estimator minimizesand takes ${\hat{m}}_n(x;p,h)={\hat{b}}_0$. Put $X_i^T=Q((x_i-x)/h{)}^T$, $W_{ii}=K_h(x_i-x)$, and rescale $b_j$ by $h^j$. If $X^{T}WX$ is positive definite, weighted least squares givesDefineThen ${\hat{m}}_n=n^{-1}{\sum }_i{w}_i{Y}_i$. If $R$ has degree at most $p$, fitting the noiseless response $R(x_i)$ reproduces that polynomial exactly, and hence

For nonzero $a$, the polynomial $a^{T}Q(u)$ cannot vanish throughout $[0,1]$. Thereforeand compactness of the unit sphere makes its minimum eigenvalue ${\Lambda }_p$ positive. Choose $a(p)$ and $n_0(p)$ so that $nh\ge a(p)$ makes the supplied lower bound on ${\lambda }_0$ at least ${\Lambda }_p/4$. On the kernel support, $Q$ is bounded by a constant depending only on $p$, and $K_h\le 1/(2h)$. It follows that only $O(nh)$ weights are nonzero andPolynomial reproduction cancels the Taylor polynomial of degree $r\le p$. The Hölder remainder on $|x_i-x|\le h$ is at most $C_{p}L{h}^{\beta }$, so the squared bias is at most $C_p{L}^2{h}^{2\beta }$. Independence and $\mathrm{Var}{\epsilon }_i\le {\sigma }^2$ bound the variance by $C_p{\sigma }^2/(nh)$. Combining them uniformly in $x$ and $m$ proves