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Maximum nonparametric kernel likelihood estimation for multiplicative linear regression models

Abstract

We propose a kernel density based estimation for multiplicative linear regression models. The method proposed in this article makes use of kernel smoothing nonparametric techniques to estimate the unknown density function of model error. For the hypothesis testing of parametric components, restricted estimators under the null hypothesis and test statistics are proposed. The asymptotic properties for the estimators and test statistics are established. We illustrate our proposals through simulations and an analysis of the QSAR fish bioconcentration factor data set. Our analysis provides strong evidence that the proposed kernel density based estimator is superior than the least squares estimator and least product relative error estimator in the literature, particularly for multimodal or asymmetric or heavy-tailed distributions of the model error.

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Acknowledgements

The authors thank the editor, the associate editor, and two referees for their constructive suggestions that helped us to improve the early manuscript. Bingqing Lin’s research was supported by the National Natural Science Foundation of China (Grant No. 11701386), the Natural Science Foundation of Guangdong Province (Grant No. 2020A1515010372) and the University stability support program A of Shenzhen (Grant No. 20200813151828003). Jun Zhang’s research was supported by the Natural Science Foundation of Guangdong Province (Grant No. 2020A1515010372), and the University stability support program A of Shenzhen (Grant No. 20200813151828003). Yiping Yang is supported by the Social Science Planning Project of Chongqing (2019WT58), Natural Science Foundation of Chongqing (CSTC2020JCYJ-MSXMX006), the National Social Science Foundation of China (Grant No. 18BTJ035), Research Fund of Chongqing Technology and Business University (Grant No. 2019ZKYY119), 2018 Chongqing Statistics Postgraduate Tutor Team (YDS183002).

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Correspondence to Bingqing Lin.

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Zhang, J., Lin, B. & Yang, Y. Maximum nonparametric kernel likelihood estimation for multiplicative linear regression models. Stat Papers (2021). https://doi.org/10.1007/s00362-021-01258-9

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Keywords

  • Kernel density based estimator
  • Restricted estimator
  • Least squares estimator
  • Least product relative error

Mathematics Subject Classification

  • 62G05
  • 62G08
  • 62G20