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Bivariate Tensor-Product B-Splines in a Partly Linear Model


  • He, Xuming
  • Shi, Peide


In some applications, the mean or median response is linearly related to some variables but the relation to additional variables are not easily parameterized. Partly linear models arise naturally in such circumstances. Suppose that a random sample {(Ti, Xi, Yi),i=1, 2, ..., n} is modeled byYi=XTi[beta]0+g0(Ti)+errori, whereYiis a real-valued response,Xi[set membership, variant]RpandTiranges over a unit square, andg0is an unknown function with a certain degree of smoothness. We make use of bivariate tensor-product B-splines as an approximation of the functiong0and consider M-type regression splines by minimization of [summation operator]ni=1 [rho](Yi-XTi[beta]-gn(Ti)) for some convex function[rho]. Mean, median and quantile regressions are included in this class. We show under appropriate conditions that the parameter estimate of[beta]achieves its information bound asymptotically and the function estimate ofg0attains the optimal rate of convergence in mean squared error. Our asymptotic results generalize directly to higher dimensions (for the variableT) provided that the functiong0is sufficiently smooth. Such smoothness conditions have often been assumed in the literature, but they impose practical limitations for the application of multivariate tensor product splines in function estimation. We also discuss the implementation of B-spline approximations based on commonly used knot selection criteria together with a simulation study of both mean and median regressions of partly linear models.

Suggested Citation

  • He, Xuming & Shi, Peide, 1996. "Bivariate Tensor-Product B-Splines in a Partly Linear Model," Journal of Multivariate Analysis, Elsevier, vol. 58(2), pages 162-181, August.
  • Handle: RePEc:eee:jmvana:v:58:y:1996:i:2:p:162-181

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    Cited by:

    1. Sherwood, Ben, 2016. "Variable selection for additive partial linear quantile regression with missing covariates," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 206-223.
    2. Kagerer, Kathrin, 2013. "A short introduction to splines in least squares regression analysis," University of Regensburg Working Papers in Business, Economics and Management Information Systems 472, University of Regensburg, Department of Economics.
    3. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "A simple and general approach to fitting the discount curve under no-arbitrage constraints," Finance Research Letters, Elsevier, vol. 15(C), pages 78-84.
    4. Huang, Jianhua Z., 2003. "Asymptotics for polynomial spline regression under weak conditions," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 207-216, November.
    5. Ying Lu & Jiang Du & Zhimeng Sun, 2014. "Functional partially linear quantile regression model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(2), pages 317-332, February.
    6. Zhao, Weihua & Lian, Heng, 2017. "Quantile index coefficient model with variable selection," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 40-58.
    7. Ibacache-Pulgar, Germán & Paula, Gilberto A., 2011. "Local influence for Student-t partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1462-1478, March.
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    10. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    11. Raheem, S.M. Enayetur & Ahmed, S. Ejaz & Doksum, Kjell A., 2012. "Absolute penalty and shrinkage estimation in partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 874-891.
    12. Horowitz, Joel L. & Lee, Sokbae, 2005. "Nonparametric Estimation of an Additive Quantile Regression Model," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1238-1249, December.
    13. Cai, Zongwu & Xiao, Zhijie, 2012. "Semiparametric quantile regression estimation in dynamic models with partially varying coefficients," Journal of Econometrics, Elsevier, vol. 167(2), pages 413-425.
    14. Sobotka, Fabian & Kneib, Thomas, 2012. "Geoadditive expectile regression," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 755-767.
    15. Qin, Guoyou & Zhang, Jiajia & Zhu, Zhongyi, 2016. "Simultaneous mean and covariance estimation of partially linear models for longitudinal data with missing responses and covariate measurement error," Computational Statistics & Data Analysis, Elsevier, vol. 96(C), pages 24-39.
    16. Tang Qingguo, 2009. "Asymptotic normality of M-estimators in a semiparametric model with longitudinal data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 69(1), pages 55-67, January.
    17. Harding, Matthew & Lamarche, Carlos, 2013. "Penalized Quantile Regression with Semiparametric Correlated Effects: Applications with Heterogeneous Preferences," IZA Discussion Papers 7741, Institute for the Study of Labor (IZA).
    18. He, Xuming & Liang, Hua, 1997. "Quantile regression estimates for a class of linear and partially linear errors-in-variables models," SFB 373 Discussion Papers 1997,103, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    19. Fengler, Matthias R. & Hin, Lin-Yee, 2015. "Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints," Journal of Econometrics, Elsevier, vol. 184(2), pages 242-261.
    20. Ganggang Xu & Suojin Wang & Jianhua Z. Huang, 2014. "Focused information criterion and model averaging based on weighted composite quantile regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(2), pages 365-381, June.
    21. Wong, Heung & Zhang, Riquan & Ip, Wai-cheung & Li, Guoying, 2008. "Functional-coefficient partially linear regression model," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 278-305, February.


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