 Solution

PutThe degree-$p$ local polynomial estimator is ${\hat{m}}_n(x)={\hat{b}}_0$, whereDefine the local polynomial Gram matrixAssume that it is positive definite, and writeThe normal equations then giveThus the estimator is a linear estimator in nonparametric regression, and the displayed $w_{p,i}$ are its effective kernel weights.

If $R$ is a polynomial of degree at most $p$, then $R(x+hu)=c^T{r}_p(u)$ for some $c$. Feeding $Y_i=R(x_i)$ into the weighted least-squares problem gives the exact zero-residual fit $c$, whose intercept is $R(x)$. Hence the polynomial reproduction property of local polynomial regression is

Let ${\beta }_0=\lceil \beta \rceil -1$. The Hölder class $\mathcal{H}(\beta ,L)$ consists of functions with derivatives through order ${\beta }_0$ bounded by $L$ andThe subclass $\tilde{\mathcal{H}}(\beta ,L)$ additionally makes every derivative of order below ${\beta }_0$ $L$-Lipschitz.

Suppose $p<{\beta }_0$. Taylor's theorem and that additional Lipschitz condition give, for the degree-$p$ Taylor polynomial $T_p$ at $x$,On the support of $K_i$, $|u_i|\le 1$, soFor the regular design $x_i=i/n$, at most $2nh+1\le 4nh$ points satisfy $|x_i-x|\le h$ when $h\ge 1/(2n)$. Polynomial reproduction cancels $T_p$, and thereforeThus the universal exponent in the question is $\gamma =1$, with the displayed choice of $C({\lambda }_0,p,L,K)$.