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Optimal variance estimation based on lagged second-order difference in nonparametric regression

Author

Listed:
  • WenWu Wang

    (Shandong University)

  • Lu Lin

    (Shandong University)

  • Li Yu

    (China Re Asset Management Company LTD)

Abstract

Differenced estimators of variance bypass the estimation of regression function and thus are simple to calculate. However, there exist two problems: most differenced estimators do not achieve the asymptotic optimal rate for the mean square error; for finite samples the estimation bias is also important and not further considered. In this paper, we estimate the variance as the intercept in a linear regression with the lagged Gasser-type variance estimator as dependent variable. For the equidistant design, our estimator is not only $$n^{1/2}$$ n 1 / 2 -consistent and asymptotically normal, but also achieves the optimal bound in terms of estimation variance with less asymptotic bias. Simulation studies show that our estimator has less mean square error than some existing differenced estimators, especially in the cases of immense oscillation of regression function and small-sized sample.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:compst:v:32:y:2017:i:3:d:10.1007_s00180-016-0666-2
    DOI: 10.1007/s00180-016-0666-2
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    References listed on IDEAS

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    1. Tiejun Tong & Yuedong Wang, 2005. "Estimating residual variance in nonparametric regression using least squares," Biometrika, Biometrika Trust, vol. 92(4), pages 821-830, December.
    2. 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.
    3. Axel Munk & Nicolai Bissantz & Thorsten Wagner & Gudrun Freitag, 2005. "On difference‐based variance estimation in nonparametric regression when the covariate is high dimensional," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 19-41, February.
    4. Jichang Du & Anton Schick, 2009. "A covariate-matched estimator of the error variance in nonparametric regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 263-285.
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    Cited by:

    1. 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.
    2. 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.
    3. 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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