= Solution
At a target $x$, the degree-$p$ <local polynomial regression> estimate minimizes
$$
\sum_{i=1}^nK\left(\frac{x_i-x}{h}\right)
\left\{Y_i-\sum_{j=0}^pb_j(x_i-x)^j\right\}^2,
$$
and $\widehat m_n(x;p,h,K)=\widehat b_0$. Let $R_x$ have row
$(1,x_i-x,\ldots,(x_i-x)^p)$ and let $W_x$ be diagonal with the kernel weights. Assume the <local polynomial Gram matrix> $R_x^TW_xR_x$ is positive definite. The weighted normal equations give
$$
\widehat m_n(x;p,h,K)
=e_1^T(R_x^TW_xR_x)^{-1}R_x^TW_xY,
$$
so it is a <linear estimator in nonparametric regression>.
Define
$$
s_r(x)=\frac1{nh}\sum_{i=1}^n(x_i-x)^r
K\left(\frac{x_i-x}{h}\right).
$$
Inverting the two-by-two Gram matrix for $p=1$ yields
$$
\widehat m_n(x;1,h,K)
=\frac1{nh}\sum_{i=1}^n
\frac{s_2(x)-s_1(x)(x_i-x)}
{s_2(x)s_0(x)-s_1(x)^2}
K\left(\frac{x_i-x}{h}\right)Y_i.
$$
Now put $M=\min\{n,\lfloor nh\rfloor\}$. At $x=0$ and for the uniform kernel,
$$
s_r=\frac1{2nh\,n^r}\sum_{i=1}^Mi^r.
$$
For the usual bandwidth range $0<h\leq1$, $M=\lfloor nh\rfloor$. The elementary power-sum formulas and $|M-nh|\leq1$ show, for $r=0,1,2,3$, that
$$
\left|s_r-\frac{h^r}{2(r+1)}\right|
\leq C_r\frac{h^r}{nh}
$$
The limiting moment determinant is
$$
\frac12\frac{h^2}{6}-\left(\frac h4\right)^2
=\frac{h^2}{48}.
$$
For $nh\geq32$, the displayed errors can consume at most a fixed fraction of this value, so
$$
s_2s_0-s_1^2\geq ch^2
$$
for a universal $c>0$ and $0<h\leq1$.
As printed, the claim “for all $h>0$” cannot hold for this design: if $h>1$, all $n$ points receive weight $1/2$, each $s_r=O(h^{-1})$, and the determinant is $O(h^{-2})$, not bounded below by a positive multiple of $h^2$. The standard bandwidth restriction $h\leq1$ is therefore necessary for that intermediate assertion. The final bias bound remains harmless for $h>1$, since $h^2\geq1$ and the finite design response means are uniformly bounded.
The <polynomial reproduction property of local polynomial regression> makes the local-linear weights reproduce both $1$ and $u$. Write
$$
e^{x_i}=1+x_i+R_i,\qquad |R_i|\leq Cx_i^2
$$
for $0\leq x_i\leq\min(h,1)$. The reproduced terms have zero bias. Using $s_r=O(h^r)$, the numerator contributed by the remainders is bounded by
$$
C\{s_2s_2+s_1s_3\}=O(h^4).
$$
Division by the determinant lower bound gives
$$
\left|\operatorname{Bias}\widehat m_n(0;1,h,K)\right|\leq Ch^2.
$$
The degree-zero <Nadaraya–Watson estimator> reproduces constants but not linear functions. The term $x_i$ therefore remains, and its boundary bias is bounded by $Ch$; this first-order boundary bias is the improvement that local linear fitting removes.
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