On the consistency of the P–C estimator in a nonparametric regression model

Regular Article


In this paper, we investigate the nonparametric regression model based on extended negatively dependent errors. Some consistency results for the estimator of the regression function g(x) are presented, including the rates of strong consistency and complete consistency, and the mean convergence. The results obtained in this paper improve and extend the corresponding ones of Yang and Wang (Acta Math Appl Sin 22(4):522–530, 1999) and Priestley and Chao (J R Stat Soc B 34:385–392, 1972). Finally, we present a numerical simulation study to verify the validity of the results established here.


Extended negatively dependent random variables P–C estimator Nonparametric regression Convergence rate Consistency 

Mathematics Subject Classification




The authors are most grateful to the Editor-in-Chief and anonymous referees for a careful reading of the manuscript and for making valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11671012, 11501004, 11501005), the Natural Science Foundation of Anhui Province (1508085J06) and The Key Projects for Academic Talent of Anhui Province (gxbjZD2016005).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Yi Wu
    • 1
  • Xuejun Wang
    • 1
  • Narayanaswamy Balakrishnan
    • 2
  1. 1.School of Mathematical SciencesAnhui UniversityHefeiPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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