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Estimating residual variance in nonparametric regression using least squares

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  • Tiejun Tong
  • Yuedong Wang

Abstract

We propose a new estimator for the error variance in a nonparametric regression model. We estimate the error variance as the intercept in a simple linear regression model with squared differences of paired observations as the dependent variable and squared distances between the paired covariates as the regressor. For the special case of a one-dimensional domain with equally spaced design points, we show that our method reaches an asymptotic optimal rate which is not achieved by some existing methods. We conduct extensive simulations to evaluate finite-sample performance of our method and compare it with existing methods. Our method can be extended to nonparametric regression models with multivariate functions defined on arbitrary subsets of normed spaces, possibly observed on unequally spaced or clustered designed points. Copyright 2005, Oxford University Press.

Suggested Citation

  • Tiejun Tong & Yuedong Wang, 2005. "Estimating residual variance in nonparametric regression using least squares," Biometrika, Biometrika Trust, vol. 92(4), pages 821-830, December.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:4:p:821-830
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    File URL: http://hdl.handle.net/10.1093/biomet/92.4.821
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    Citations

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    Cited by:

    1. Martin Meermeyer, 2015. "Weighted linear regression models with fixed weights and spherical disturbances," Computational Statistics, Springer, vol. 30(4), pages 929-955, December.
    2. Lu Lin & Feng Li, 2008. "Stable and bias-corrected estimation for nonparametric regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(4), pages 283-303.
    3. Peter Hall & Joel L. Horowitz, 2012. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers 14/12, Institute for Fiscal Studies.
    4. Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers CWP29/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Mendez, Guillermo & Lohr, Sharon, 2011. "Estimating residual variance in random forest regression," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2937-2950, November.
    6. Peter Hall & Joel L. Horowitz, 2012. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers CWP14/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Wang, WenWu & Yu, Ping, 2017. "Asymptotically optimal differenced estimators of error variance in nonparametric regression," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 125-143.
    8. Zhijian Li & Wei Lin, 2020. "Efficient error variance estimation in non‐parametric regression," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 62(4), pages 467-484, December.
    9. Germán Aneiros & Nengxiang Ling & Philippe Vieu, 2015. "Error variance estimation in semi-functional partially linear regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 316-330, September.
    10. Lu Lin & Feng Li, 2023. "Global debiased DC estimations for biased estimators via pro forma regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 726-758, June.
    11. Liitiäinen, Elia & Corona, Francesco & Lendasse, Amaury, 2010. "Residual variance estimation using a nearest neighbor statistic," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 811-823, April.
    12. WenWu Wang & Lu Lin & Li Yu, 2017. "Optimal variance estimation based on lagged second-order difference in nonparametric regression," Computational Statistics, Springer, vol. 32(3), pages 1047-1063, September.
    13. Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers 29/13, Institute for Fiscal Studies.
    14. Yuejin Zhou & Yebin Cheng & Wenlin Dai & Tiejun Tong, 2018. "Optimal difference-based estimation for partially linear models," Computational Statistics, Springer, vol. 33(2), pages 863-885, June.
    15. Eckhard Liebscher, 2012. "Model checks for parametric regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 132-155, March.
    16. Jingxin Zhao & Heng Peng & Tao Huang, 2018. "Variance estimation for semiparametric regression models by local averaging," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 453-476, June.
    17. Lu Lin & Xia Cui & Lixing Zhu, 2009. "An Adaptive Two‐stage Estimation Method for Additive Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 248-269, June.
    18. Hui-Ling Lin & Zhouping Li & Dongliang Wang & Yichuan Zhao, 2017. "Jackknife empirical likelihood for the error variance in linear models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 151-166, April.
    19. Wenlin Dai & Tiejun Tong, 2014. "Variance estimation in nonparametric regression with jump discontinuities," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(3), pages 530-545, March.

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