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Optimal difference-based estimation for partially linear models

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Abstract

Difference-based methods have attracted increasing attention for analyzing partially linear models in the recent literature. In this paper, we first propose to solve the optimal sequence selection problem in difference-based estimation for the linear component. To achieve the goal, a family of new sequences and a cross-validation method for selecting the adaptive sequence are proposed. We demonstrate that the existing sequences are only extreme cases in the proposed family. Secondly, we propose a new estimator for the residual variance by fitting a linear regression method to some difference-based estimators. Our proposed estimator achieves the asymptotic optimal rate of mean squared error. Simulation studies also demonstrate that our proposed estimator performs better than the existing estimator, especially when the sample size is small and the nonparametric function is rough.

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References

  • Akdeniz F, Duran E (2013) New difference-based estimator of parameters in semiparametric regression models. J Stat Comput Simul 83:810–824

    Article  MathSciNet  MATH  Google Scholar 

  • Aneiros G, Ling N, Vieu P (2015) Error variance estimation in semi-functional partially linear regression models. J Nonparametr Stat 27:316–330

    Article  MathSciNet  MATH  Google Scholar 

  • Chen H, Shiau J (1991) A two-stage spline smoothing method for partially linear models. J Stat Plan Inference 27:187–201

    Article  MathSciNet  MATH  Google Scholar 

  • Cuzick J (1992) Semiparametric additive regression. J Roy Stat Soc B 54:831–843

    MathSciNet  MATH  Google Scholar 

  • Dai W, Ma Y, Tong T, Zhu L (2015) Difference-based variance estimation in nonparametric regression with repeated measurement data. J Stat Plan Inference 163:1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Dai W, Tong T, Genton M (2016) Optimal estimation of derivatives in nonparametric regression. J Mach Learn Res 17:1–25

    MathSciNet  MATH  Google Scholar 

  • Dai W, Tong T, Zhu L (2017) On the choice of difference sequence in a unified framework for variance estimation in nonparametric regression. Stat Sci 32:455–468

    Article  MathSciNet  Google Scholar 

  • Dette H, Munk A, Wagner T (1998) Estimating the variance in nonparametric regression—what is a reasonable choice? J Roy Stat Soc B 60:751–764

    Article  MathSciNet  MATH  Google Scholar 

  • Eubank R, Kambour E, Kim J, Klipple K, Reese C, Schimek M (1998) Estimation in partially linear models. Comput Stat Data Anal 29:27–34

    Article  MathSciNet  MATH  Google Scholar 

  • Fan J, Huang T (2005) Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11:1031–1057

    Article  MathSciNet  MATH  Google Scholar 

  • Gasser T, Sroka L, Jennen-Steinmetz C (1986) Residual variance and residual pattern in nonlinear regression. Biometrika 73:625–633

    Article  MathSciNet  MATH  Google Scholar 

  • Hall P, Kay JW, Titterington DM (1990) Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77:521–528

    Article  MathSciNet  MATH  Google Scholar 

  • Hall P, Keilegom IV (2003) Using difference-based methods for inference in nonparametric regression with time series errors. J Roy Stat Soc B 65:443–456

    Article  MathSciNet  MATH  Google Scholar 

  • Hardle W, Liang H, Gao J (2000) Partially linear models. Physika Verlag, Heidelberg

    Book  MATH  Google Scholar 

  • He H, Tang W, Zuo G (2014) Statistical inference in the partial inear models with the doubles moothing local linear regression method. J Stat Plan Inference 146:102–112

    Article  Google Scholar 

  • Hu H, Zhang Y, Pan X (2016) Asymptotic normality of dhd estimators in a partially linear model. Stat Pap 57:567–587

    Article  MathSciNet  MATH  Google Scholar 

  • Levine M (2015) Minimax rate of convergence for an estimator of the functional component in a semiparametric multivariate partially linear model. J Multivar Anal 140:283–290

    Article  MathSciNet  MATH  Google Scholar 

  • Liu Q, Zhao G (2012) A comparison of estimation methods for partially linear models. Int J Innov Manag Inf Prod 3:38–42

    Google Scholar 

  • Lokshin M (2006) Difference-based semiparametric estimation of partial linear regression models. Stata J 6:377–383

    Google Scholar 

  • Rice JA (1984) Bandwidth choice for nonparametric regression. Ann Stat 12:1215–1230

    Article  MathSciNet  MATH  Google Scholar 

  • Schott JR (1997) Matrix analysis for statistics. Wiley, New York

    MATH  Google Scholar 

  • Severini T, Wong W (1992) Generalized profile likelihhood and conditional parametric models. Ann Stat 20:1768–1802

    Article  MATH  Google Scholar 

  • Spechman P (1988) Kernel smoothing in partial linear models. J Roy Stat Soc B 50:413–436

    MathSciNet  Google Scholar 

  • Tabakan G (2013) Performance of the difference-based estimators in partially linear models. Statistics 47:329–347

    Article  MathSciNet  MATH  Google Scholar 

  • Tong T, Ma Y, Wang Y (2013) Optimal variance estimation without estimating the meam function. Bernoulli 19:1839–1854

    Article  MathSciNet  MATH  Google Scholar 

  • Tong T, Wang Y (2005) Estimating residual variance in nonparametric regression using least squares. Biometrika 92:821–830

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Brown L, Cai T (2011) A difference based approach to the semiparametric partial linear model. Electron J Stat 5:619–641

    Article  MathSciNet  MATH  Google Scholar 

  • Wang W, Lin L (2015) Derivative estimation based on difference sequence via locally weighted least squares regession. J Mach Learn Res 16:2617–2641

    MathSciNet  MATH  Google Scholar 

  • Wang W, Lin L, Yu L (2017) Optimal variance estimation based on lagged second-order difference in nonparametric regression. Comput Stat 32:1047–1063

    Article  MathSciNet  MATH  Google Scholar 

  • Whittle P (1964) On the convergence to normality of quadratic forms in independent variables. Teor. Verojatnost. i Primenen 9:113–118

    MathSciNet  MATH  Google Scholar 

  • Yatchew A (1997) An elementary estimator of the partial linear model. Econ Lett 57:135–143

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao H, You J (2011) Difference based estimation for partially linear regression models with measurement errors. J Multivar Anal 102:1321–1338

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou Y, Cheng Y, Wang L, Tong T (2015) Optimal difference-based variance estimation in heteroscedastic nonparametric regression. Stat Sin 25:1377–1397

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

Yuejin Zhou’s research was supported in part by the Natural Science Foundation of Anhui Grant (No. KJ2017A087), and the National Natural Science Foundation of China Grant (No. 61472003). Yebin Cheng’s research was supported in part by the National Natural Science Foundation of China Grant (No. 11271241). Tiejun Tong’s research was supported in part by the Hong Kong Baptist University Grants FRG1/16-17/018 and FRG2/16-17/074, and the National Natural Science Foundation of China Grant (No. 11671338).

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Authors and Affiliations

  1. School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, China

    Yuejin Zhou

  2. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, China

    Yuejin Zhou

  3. Glorious Sun School of Business and Management, Donghua University, Shanghai, China

    Yebin Cheng

  4. CEMSE Division, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia

    Wenlin Dai

  5. Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China

    Tiejun Tong

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  1. Yuejin Zhou
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  2. Yebin Cheng
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  3. Wenlin Dai
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  4. Tiejun Tong
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Correspondence to Yuejin Zhou.

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Zhou, Y., Cheng, Y., Dai, W. et al. Optimal difference-based estimation for partially linear models. Comput Stat 33, 863–885 (2018). https://doi.org/10.1007/s00180-017-0786-3

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