The general condition is that $Y_i = r(X_i) + \epsilon_i$, and we want to estimate $r$ using Nadaraya–Watson kernel regression.
We additionally assume $r\colon [0,1] \to \mathbb{R}$ is lipschitz, so $|r(x)-r(y)| \leq L|x-y|$ for all $x,y \in [0,1]$.
We want to find the maximum bias of NW estimator in this case.
We know that approximate bias is $\frac{h^2}{2}\sigma_K^2 \left[ r''(x) + \frac{r'(x) f'(x)}{f(x)} \right]$. And since $r$ Lipschitz, $|r(x) - r(x+h)| \leq L|h|$, and hence, $r'(x) \leq L$.
But just replacing $r'(x)$ by $L$ in the equation seems like it's missing something.
What am I missing?
