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On complete consistency for the weighted estimator of nonparametric regression models

  • Rui Zhang
  • Yi Wu
  • Weifeng Xu
  • Xuejun WangEmail author
Original Paper
  • 12 Downloads

Abstract

In this paper, we consider the following nonparametric regression model:
$$\begin{aligned} Y_{ni} = f\left( x_{ni}\right) + \varepsilon _{ni},\quad i = 1, 2, \ldots , n,\quad n\ge 1, \end{aligned}$$
where \(x_{ni}\) are known fixed design points from A, where \(A\subset {\mathbb {R}}^d\) is a given compact set for some \(d\ge 1\), \(f(\cdot )\) is an unknown regression function defined on A and \(\varepsilon _{ni}\) are random errors, which are assumed to widely orthant dependent (WOD, for short). Firstly, a general result on complete convergence for partial sums of WOD random variables is obtained, which has some interest itself. Based on some mild conditions and the complete convergence result that we established, we further establish the complete consistency of the weighted estimator in the nonparametric regression model, which improves the corresponding one of Wang et al. (TEST 20:607–629, 2014). As an application, the complete consistency of the nearest neighbor estimator is obtained. Finally we provide a numerical simulation to verify the validity of our result.

Keywords

Widely orthant dependent random variables Weighted estimator Nonparametric regression model Complete consistency 

Mathematics Subject Classification

62G05 

Notes

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Wendian CollegeAnhui UniversityHefeiPeople’s Republic of China
  2. 2.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China

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