Let $Y\_t \in \mathbb{R}$ be a response variable and $X\_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X\_1, Y\_1), \cdots, (X\_n, Y\_n)$.
\begin{equation}
 Y\_{t} = \mu(X\_{t})+\sigma(X\_{t})\varepsilon\_{t}, \quad t\in\mathbb{Z}
\end{equation}
Then the NW estimator for $\mu(\cdot)$ and $\sigma^2(\cdot)$ are
\begin{equation\*}
 \hat{\mu}(x) = \frac{1}{n \bar{w}(x; h)} \sum\_{t = 1}^n w(x ; t, h) Y\_t,
\end{equation\*}
and
\begin{equation\*}
 \hat\sigma^2(x) = \frac{1}{n \bar{w}(x; h)} \sum\_{t = 1}^n w (x ; t, h) \big( Y\_t - \hat{\mu} (x)\big)^2,
\end{equation\*}
respectively, where
\begin{equation\*}
 \bar{w}(x ; h) = \frac{1}{n} \sum\_{t = 1}^n w(x ; t, h), \qquad w(x ; t, h) = \frac{1}{h^d} H\left( \frac{X\_t - x}{h}\right)
\end{equation\*}
is the $d$-dimensional kernel density estimator of $f(x)$, the marginal density of $\{X\_t\}$.


Now I want to know the properties of the two estimators, like the asymptotic expansion of $\hat{\mu}(x) - \mu(x)$ and $\hat{\sigma}^2(x) - \sigma^2(x)$.


There does exist some articles regarding the asymptotic normality for $\hat{\sigma}^2(x) - \sigma^2(x)$ in the setting of the local linear method, see [Fan and Qao](http://eprints.lse.ac.uk/6635/1/Efficient_estimation_of_conditional_variance_functions_in_stochastic_regression%28LSERO%29.pdf). However, I cannot find any useful literature for the asymptotic expansion of $\hat{\mu}(x) - \mu(x)$ and $\hat{\sigma}^2(x) - \sigma^2(x)$ in the setting of NW estimators.


Could anyone help me? Thanks so much!



The nonparametric estimation in generalized regression model
The asymptotic expression for $(\hat{m} - m)(x)$ is presented in the Appendix of [Fan and Qao](http://eprints.lse.ac.uk/6635/1/Efficient_estimation_of_conditional_variance_functions_in_stochastic_regression%28LSERO%29.pdf). So this question is meaningless.


