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

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  • He, Xuming
  • Shi, Peide
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    Abstract

    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 58 (1996)
    Issue (Month): 2 (August)
    Pages: 162-181

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    Handle: RePEc:eee:jmvana:v:58:y:1996:i:2:p:162-181

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    Keywords: B-spline functions rate of convergence mean regression median regression M-estimator partly linear model regression quantile;

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    Cited by:
    1. Huang, Jianhua Z., 2003. "Asymptotics for polynomial spline regression under weak conditions," Statistics & Probability Letters, Elsevier, Elsevier, vol. 65(3), pages 207-216, November.
    2. Wong, Heung & Zhang, Riquan & Ip, Wai-cheung & Li, Guoying, 2008. "Functional-coefficient partially linear regression model," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 99(2), pages 278-305, February.
    3. 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.
    4. Joel Horowitz & Sokbae 'Simon' Lee, 2004. "Nonparametric estimation of an additive quantile regression model," CeMMAP working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies CWP07/04, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Fengler, Matthias R. & Hin, Lin-Yee, 2014. "A simple and general approach to fitting the discount curve under no-arbitrage constraints," Economics Working Paper Series 1423, University of St. Gallen, School of Economics and Political Science.
    6. Zongwu Cai & Zhijie Xiao, 2010. "Semiparametric Quantile Regression Estimation in Dynamic Models with Partially Varying Coefficients," Boston College Working Papers in Economics, Boston College Department of Economics 761, Boston College Department of Economics.
    7. 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, Elsevier, vol. 56(4), pages 874-891.
    8. 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.
    9. Tang Qingguo, 2009. "Asymptotic normality of M-estimators in a semiparametric model with longitudinal data," Metrika, Springer, Springer, vol. 69(1), pages 55-67, January.
    10. Ying Lu & Jiang Du & Zhimeng Sun, 2014. "Functional partially linear quantile regression model," Metrika, Springer, Springer, vol. 77(2), pages 317-332, February.
    11. repec:wyi:journl:002114 is not listed on IDEAS
    12. 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).
    13. Sobotka, Fabian & Kneib, Thomas, 2012. "Geoadditive expectile regression," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 56(4), pages 755-767.

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