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Estimating the variance in nonparametric regression—what is a reasonable choice?

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  • H. Dette
  • A. Munk
  • T. Wagner

Abstract

The exact mean‐squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titterington's optimal difference‐based estimators and a class of ordinary difference‐based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen‐Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function. It is shown that in many situations ordinary difference‐based estimators are more appropriate for estimating the variance, because they control the bias much better and hence have a much better overall performance. It is also demonstrated that Rice's estimator does not always behave well. Data‐driven guidelines are given to select the estimator with the smallest MSE.

Suggested Citation

  • H. Dette & A. Munk & T. Wagner, 1998. "Estimating the variance in nonparametric regression—what is a reasonable choice?," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 751-764.
    • Handle: RePEc:bla:jorssb:v:60:y:1998:i:4:p:751-764
    DOI: 10.1111/1467-9868.00152
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    Cited by:

    1. 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.
    2. Einmahl, J.H.J. & van Keilegom, I., 2006. "Tests for Independence in Nonparametric Regression," Other publications TiSEM 0c6f2c43-aa7d-45c1-9d43-7, Tilburg University, School of Economics and Management.
    3. 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.
    4. Pierre Bellec, 2015. "Optimal bounds for aggregation of affine estimators," Working Papers 2015-06, Center for Research in Economics and Statistics.
    5. Einmahl, J.H.J. & van Keilegom, I., 2004. "Goodness-of-fit Tests in Nonparametric Regression," Other publications TiSEM 44e08f75-b35d-424e-b33e-0, Tilburg University, School of Economics and Management.
    6. Mathias Lindholm & Felix Wahl, 2020. "On the variance parameter estimator in general linear models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(2), pages 243-254, February.
    7. Holger Dette & Theresa Eckle & Mathias Vetter, 2020. "Multiscale change point detection for dependent data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1243-1274, December.
    8. 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.
    9. Ieva Axt & Roland Fried, 2020. "On variance estimation under shifts in the mean," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(3), pages 417-457, September.

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