0
$\begingroup$

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. 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!

$\endgroup$
0
$\begingroup$

The asymptotic expression for $(\hat{m} - m)(x)$ is presented in the Appendix of Fan and Qao. So this question is meaningless.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.