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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 $\hat{\sigma}^2(x) - \sigma^2(x)$ when $n \rightarrow \infty$.

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!

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  • $\begingroup$ What makes asymptotics relevant, since you would be applying the method to finite samples? $\endgroup$ – Frank Harrell Nov 5 '20 at 13:04
  • $\begingroup$ @FrankHarrell Yeah. I forgot adding the condition that $n \rightarrow \infty$. $\endgroup$ – 香结丁 Nov 6 '20 at 2:03
  • $\begingroup$ That doesn't come close to answering my question. $\endgroup$ – Frank Harrell Nov 6 '20 at 12:59
  • $\begingroup$ @FrankHarrell Sorry... I may not get your point. Do you mean I should state the regular conditions to let the NW-estimators make sense? $\endgroup$ – 香结丁 Nov 7 '20 at 1:22
  • $\begingroup$ No. You should validate performance on realistic sample sizes. $\endgroup$ – Frank Harrell Nov 7 '20 at 13:26
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The asymptotic expression for $(\hat{m} - m)(x)$ is presented in the Appendix of Fan and Qao. So this question is meaningless.

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