# The nonparametric estimation in generalized regression model

Let $$Y_t \in \mathbb{R}$$ be a response variable and $$X_t$$ a $$d$$-dimensional explanatory variable. Assume we observe the process that $$(X_1, Y_1), \cdots, (X_n, Y_n)$$. $$$$Y_{t} = \mu(X_{t})+\sigma(X_{t})\varepsilon_{t}, \quad t\in\mathbb{Z}$$$$ Then the NW estimator for $$\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!

• What makes asymptotics relevant, since you would be applying the method to finite samples? – Frank Harrell Nov 5 '20 at 13:04
• @FrankHarrell Yeah. I forgot adding the condition that $n \rightarrow \infty$. – 香结丁 Nov 6 '20 at 2:03
• That doesn't come close to answering my question. – Frank Harrell Nov 6 '20 at 12:59
• @FrankHarrell Sorry... I may not get your point. Do you mean I should state the regular conditions to let the NW-estimators make sense? – 香结丁 Nov 7 '20 at 1:22
• No. You should validate performance on realistic sample sizes. – Frank Harrell Nov 7 '20 at 13:26

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