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Efficient semiparametric estimator for heteroscedastic partially linear models

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  • Yanyuan Ma
  • Jeng-Min Chiou
  • Naisyin Wang

Abstract

We study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function. An interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated. To avoid this, we propose a family of consistent estimators and investigate their asymptotic properties. We show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family. Building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing, we conjecture that a weighted partial-spline estimator could also be semiparametric efficient. Properties of the proposed estimators are presented through theoretical illustration and numerical simulations. Copyright 2006, Oxford University Press.

Suggested Citation

  • Yanyuan Ma & Jeng-Min Chiou & Naisyin Wang, 2006. "Efficient semiparametric estimator for heteroscedastic partially linear models," Biometrika, Biometrika Trust, vol. 93(1), pages 75-84, March.
    • Handle: RePEc:oup:biomet:v:93:y:2006:i:1:p:75-84
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    File URL: http://hdl.handle.net/10.1093/biomet/93.1.75
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    Cited by:

    1. Runze Li & Lei Nie, 2008. "Efficient Statistical Inference Procedures for Partially Nonlinear Models and their Applications," Biometrics, The International Biometric Society, vol. 64(3), pages 904-911, September.
    2. Lai, Peng & Wang, Qihua, 2014. "Semiparametric efficient estimation for partially linear single-index models with responses missing at random," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 33-50.
    3. Weibin Mo & Yufeng Liu, 2022. "Efficient learning of optimal individualized treatment rules for heteroscedastic or misspecified treatment‐free effect models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 440-472, April.
    4. Hohsuk Noh & Seong J. Yang, 2020. "Comparing Groups of Decision-Making Units in Efficiency Based on Semiparametric Regression," Mathematics, MDPI, vol. 8(2), pages 1-16, February.
    5. Mijeong Kim & Yanyuan Ma, 2019. "Semiparametric efficient estimators in heteroscedastic error models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 1-28, February.
    6. Jeremiah Jones & Ashkan Ertefaie & Susan M. Shortreed, 2023. "Rejoinder to “Reader reaction to ‘Outcome‐adaptive Lasso: Variable selection for causal inference’ by Shortreed and Ertefaie (2017)”," Biometrics, The International Biometric Society, vol. 79(1), pages 521-525, March.
    7. Lai, Peng & Wang, Qihua & Zhou, Xiao-Hua, 2014. "Variable selection and semiparametric efficient estimation for the heteroscedastic partially linear single-index model," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 241-256.
    8. Tang, Niansheng & Xia, Linli & Yan, Xiaodong, 2019. "Feature screening in ultrahigh-dimensional partially linear models with missing responses at random," Computational Statistics & Data Analysis, Elsevier, vol. 133(C), pages 208-227.
    9. Otsu, Taisuke, 2007. "Penalized empirical likelihood estimation of semiparametric models," Journal of Multivariate Analysis, Elsevier, vol. 98(10), pages 1923-1954, November.
    10. Kun Chen & Yanyuan Ma, 2017. "Analysis of Double Single Index Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 1-20, March.
    11. Zhou, Ling & Lin, Huazhen & Chen, Kani & Liang, Hua, 2019. "Efficient estimation and computation of parameters and nonparametric functions in generalized semi/non-parametric regression models," Journal of Econometrics, Elsevier, vol. 213(2), pages 593-607.
    12. Peirong Xu & Jun Zhang & Xingfang Huang & Tao Wang, 2016. "Efficient estimation for marginal generalized partially linear single-index models with longitudinal data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 413-431, September.
    13. Harrison H. Li & Art B. Owen, 2023. "Double machine learning and design in batch adaptive experiments," Papers 2309.15297, arXiv.org.
    14. Wang, Xiaoguang & Lu, Dawei & Song, Lixin, 2013. "Statistical inference for partially linear stochastic models with heteroscedastic errors," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 150-160.
    15. Lu, Xuewen, 2009. "Empirical likelihood for heteroscedastic partially linear models," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 387-396, March.
    16. Jianglin Fang & Wanrong Liu & Xuewen Lu, 2018. "Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(3), pages 255-281, April.
    17. Yixin Fang & Heng Lian & Hua Liang, 2018. "A generalized partially linear framework for variance functions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1147-1175, October.
    18. Tianfa Xie & Zhihua Sun & Liuquan Sun, 2012. "A consistent model specification test for a partial linear model with covariates missing at random," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 841-856, December.
    19. Mijeong Kim & Yanyuan Ma, 2012. "The efficiency of the second-order nonlinear least squares estimator and its extension," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 751-764, August.
    20. Long, Wei & Ouyang, Min & Shang, Ying, 2013. "Efficient estimation of partially linear varying coefficient models," Economics Letters, Elsevier, vol. 121(1), pages 79-81.

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