The Berry–Esseen-type bound for the G-M estimator in a nonparametric regression model with α-mixing errors

Abstract

Consider the fixed design nonparametric regression model: $Y_i=g(x_i)+{\epsilon }_i,i=1,2,\dots ,n,n\ge 1$, where $x_i$ are known fixed design points from $[0,1]$ satisfying $0=x_0\le x_1\le \cdots \le x_{n-1}\le x_n=1$ and $g(x)$ is an unknown function defined on the closed interval $[0,1]$, $Y_i$ are the response variables, and ${\epsilon }_i$ are zero mean α-mixing random variables with $\alpha (n)=O(n^{-\lambda })\mathrm{for}\mathrm{some}\lambda >(2+\delta )(1+\delta )/\delta$, where $\delta >0$. We derive the Berry–Esseen-type bound for the G-M estimator of $g(\cdot )$ under some mild conditions, which approximates to $O\left(n^{-\frac{1}{4}}\right)$ provided that weights and moments are appropriate. Our results generalize some known ones in the literature. In addition, a simulation study is given to illustrate the validity of the theoretical result.

Keywords:

• Nonparametric regression model
• α-mixing random variables
• G-M estimator
• Berry–Esseen-type bound

Mathematical Subject Classifications:

• 62G20
• 62E20

Acknowledgements

The authors would like to express their deep thanks to the Editor and anonymous referees for carefully reading the manuscript and for their useful comments, which greatly helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11871072, 12001105], the Natural Science Foundation of Anhui Province [2108085MA06, 1908085QA01, 1908085QA07], and the Postdoctoral Science Foundation of China [2019M660156].