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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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    1. repec:spr:compst:v:33:y:2018:i:2:d:10.1007_s00180-017-0786-3 is not listed on IDEAS
    2. 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.
    3. Martin Meermeyer, 2015. "Weighted linear regression models with fixed weights and spherical disturbances," Computational Statistics, Springer, vol. 30(4), pages 929-955, December.
    4. repec:spr:testjl:v:27:y:2018:i:2:d:10.1007_s11749-017-0553-3 is not listed on IDEAS
    5. 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.
    6. 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.
    7. 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.
    8. Mendez, Guillermo & Lohr, Sharon, 2011. "Estimating residual variance in random forest regression," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2937-2950, November.
    9. repec:spr:compst:v:32:y:2017:i:3:d:10.1007_s00180-016-0666-2 is not listed on IDEAS
    10. 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.
    11. 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.

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