Advanced Search
MyIDEAS: Login to save this article or follow this journal

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


Author Info

  • He, Xuming
  • Shi, Peide
Registered author(s):


    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.

    Download Info

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
    File URL:
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

    Bibliographic Info

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

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

    as in new window
    Handle: RePEc:eee:jmvana:v:58:y:1996:i:2:p:162-181

    Contact details of provider:
    Web page:

    Order Information:

    Related research

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


    No references listed on IDEAS
    You can help add them by filling out this form.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as in new window

    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.


    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.


    Access and download statistics


    When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:58:y:1996:i:2:p:162-181. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Zhang, Lei).

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.