The asymptotic normality of the linear weighted estimator in nonparametric regression models

ABSTRACT

Consider the following nonparametric regression model: $$Y_{ni}=g\left(x_{ni}\right)+{\epsilon }_{ni},i=1,2,\dots ,n,n\ge 1,$$where x_ni are known fixed design points from $A\subset {\mathbb{R}}^d$ for some positive integer d ⩾ 1, g( · ) is an unknown regression function defined on A and ϵ_ni are random errors. Under some suitable conditions, the asymptotic normality of the linear weighted estimator of g in the nonparametric regression model based on ρ-mixing errors is established. The key techniques used in the paper are the Rosenthal type inequality and the Bernstein’s bigblock and small-block procedure. The result obtained in the paper generalizes the corresponding ones for some dependent sequences.

KEYWORDS:

• Asymptotic normality
• Linear weighted estimator
• Nonparametric regression model
• ρ-mixing random variables.

MATHEMATICS SUBJECT CLASSIFICATION:

• MSC(2010): 62G20

Acknowledgments

The authors are grateful to the Referee for carefully reading the manuscript and for providing helpful comments and constructive criticism which enabled them to improve the paper.

Funding

Supported by the National Natural Science Foundation of China (11501004), the Natural Science Foundation of Anhui Province (1508085J06), the Provincial Natural Science Research Project of Anhui Colleges (KJ2015A018), the Doctoral Research Start-up Funds Projects of Anhui University and the Quality Engineering Project of Anhui Province (2015jyxm045).