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Error variance estimation in semi-functional partially linear regression models

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  • Germán Aneiros
  • Nengxiang Ling
  • Philippe Vieu

Abstract

This paper focuses on partially linear regression models with several real and functional covariates. The aim is to construct an estimate of the variance of the error. In our model, a real-valued response variable is explained by the sum of an unknown linear combination of the components of a multivariate random variable and an unknown transformation of a functional random variable, and the second sample moment based on residuals from a semiparametric fit is proposed for estimating the error variance. Then, the asymptotic normality and the law of the iterated logarithm of such estimator are obtained. Finally, a simulation study illustrates the finite sample behaviour of the estimator, while an application to real data shows the usefulness of the proposed methodology, more specifically for confidence region construction.

Suggested Citation

  • Germán Aneiros & Nengxiang Ling & Philippe Vieu, 2015. "Error variance estimation in semi-functional partially linear regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 316-330, September.
    • Handle: RePEc:taf:gnstxx:v:27:y:2015:i:3:p:316-330
    DOI: 10.1080/10485252.2015.1042376
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    References listed on IDEAS

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    1. Aneiros-Pérez, Germán & Vieu, Philippe, 2006. "Semi-functional partial linear regression," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1102-1110, June.
    2. Tiejun Tong & Yuedong Wang, 2005. "Estimating residual variance in nonparametric regression using least squares," Biometrika, Biometrika Trust, vol. 92(4), pages 821-830, December.
    3. Maity, Arnab & Huang, Jianhua Z., 2012. "Partially linear varying coefficient models stratified by a functional covariate," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1807-1814.
    4. Li, Qi, 2000. "Efficient Estimation of Additive Partially Linear Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 41(4), pages 1073-1092, November.
    5. Han Shang, 2014. "Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density," Computational Statistics, Springer, vol. 29(3), pages 829-848, June.
    6. Hardle, Wolfgang & LIang, Hua & Gao, Jiti, 2000. "Partially linear models," MPRA Paper 39562, University Library of Munich, Germany, revised 01 Sep 2000.
    7. Heng Lian, 2011. "Functional partial linear model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 115-128.
    8. Isabel Casas & Irene Gijbels, 2012. "Unstable volatility: the break-preserving local linear estimator," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 883-904, December.
    9. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.
    10. Gao, Jiti, 1995. "The laws of the iterated logarithm of some estimates in partly linear models," Statistics & Probability Letters, Elsevier, vol. 25(2), pages 153-162, November.
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    Citations

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    Cited by:

    1. Shuyu Meng & Zhensheng Huang, 2024. "Variable Selection in Semi-Functional Partially Linear Regression Models with Time Series Data," Mathematics, MDPI, vol. 12(17), pages 1-23, September.
    2. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    3. Ping Yu & Zhongyi Zhu & Zhongzhan Zhang, 2019. "Robust exponential squared loss-based estimation in semi-functional linear regression models," Computational Statistics, Springer, vol. 34(2), pages 503-525, June.
    4. Zhu, Hanbing & Zhang, Riquan & Yu, Zhou & Lian, Heng & Liu, Yanghui, 2019. "Estimation and testing for partially functional linear errors-in-variables models," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 296-314.
    5. Nengxiang Ling & Rui Kan & Philippe Vieu & Shuyu Meng, 2019. "Semi-functional partially linear regression model with responses missing at random," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(1), pages 39-70, January.
    6. 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.
    7. Nengxiang Ling & Lingyu Wang & Philippe Vieu, 2020. "Convergence rate of kernel regression estimation for time series data when both response and covariate are functional," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(6), pages 713-732, August.
    8. Chaouch, Mohamed, 2019. "Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 129-148.
    9. Boente, Graciela & Vahnovan, Alejandra, 2017. "Robust estimators in semi-functional partial linear regression models," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 59-84.
    10. Nengxiang Ling & Germán Aneiros & Philippe Vieu, 2020. "kNN estimation in functional partial linear modeling," Statistical Papers, Springer, vol. 61(1), pages 423-444, February.

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