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B-spline Speckman Estimator of Partially Linear Model

Received: 25 August 2019     Accepted: 31 December 2019     Published: 3 February 2020
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Abstract

The partially linear model (PLM) is one of semiparametric regression models; since it has both parametric (more than one) and nonparametric (only one) components in the same model, so this model is more flexible than the linear regression models containing only parametric components. In the literature, there are several estimators are proposed for this model; where the main difference between these estimators is the estimation method used to estimate the nonparametric component, since the parametric component is estimated by least squares method mostly. The Speckman estimator is one of the commonly used for estimating the parameters of the PLM, this estimator based on kernel smoothing approach to estimate nonparametric component in the model. According to the papers in nonparametric regression, in general, the spline smoothing approach is more efficient than kernel smoothing approach. Therefore, we suggested, in this paper, using the basis spline (B-spline) smoothing approach to estimate nonparametric component in the model instead of the kernel smoothing approach. To study the performance of the new estimator and compare it with other estimators, we conducted a Monte Carlo simulation study. The results of our simulation study confirmed that the proposed estimator was the best, because it has the lowest mean squared error.

Published in International Journal of Systems Science and Applied Mathematics (Volume 4, Issue 4)
DOI 10.11648/j.ijssam.20190404.12
Page(s) 53-59
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Kernel Smoothing, Monte Carlo Simulation, Penalized B-spline Estimation, Semiparametric Regression, Spline Smoothing

References
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[2] Heckman, N. E. (1986). Spline smoothing in a partly linear model. Journal of the Royal Statistical Society: Series B (Methodological), 48 (2), 244-248.
[3] Rice, J. (1986). Convergence rates for partially splined models. Statistics & probability letters, 4 (4), 203-208.
[4] Chen, H., & Shiau, J. J. H. (1991). A two-stage spline smoothing method for partially linear models. Journal of Statistical Planning and Inference, 27 (2), 187-201.
[5] Abonazel, M. R., & Gad, A. A. E. (2018). Robust partial residuals estimation in semiparametric partially linear model. Communications in Statistics-Simulation and Computation, 1-14.
[6] Abonazel, M. R., Helmy, N. & Azazy, A. (2019). The Performance of Speckman Estimation for Partially Linear Model using Kernel and Spline Smoothing Approaches. International Journal of Mathematical Archive, 10 (6):10-18.
[7] Robinson, P. M. (1988), Root-N-consistent semiparametric regression. Econometrica, 56 (4), 931–54.
[8] Speckman, P. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society: Series B (Methodological), 50 (3), 413-436.
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[11] Yatchew, A. (1997). An elementary estimator of the partial linear model. Economics Letters, 57 (2), 135–143.
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[13] Yatchew, A. (2003). Semiparametric regression for the applied econometrician. Cambridge University Press.
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[15] Henderson, D. J., & Parmeter, C. F. (2015). Single-step estimation of a partially linear model. Working Papers, University of Miami, Department of Economics. Available at: https://www.bus.miami.edu/_assets/files/repec/WP2015-01.pdf
[16] Elgohary, M. M., Abonazel, M. R., Helmy, N. M., & Azazy, A. R. (2019). New robust-ridge estimators for partially linear model. International Journal of Applied Mathematical Research, 8 (2): 46-52.
[17] Green, P. J., & Silverman, B. W. (1994). Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman and Hall/CRC.
[18] Ruppert, D., Wand, M. P., & Carroll, R. J. (2003). Semiparametric regression (No. 12). Cambridge university press.
[19] Wasserman, L. (2006). All of nonparametric statistics. Springer Science & Business Media.
[20] De Boor, C. (1978). A practical guide to splines. (Vol. 27, p. 325). New York: springer-verlag.
[21] Eilers, P. H., & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11 (2), 89-102.
[22] Eilers, P. H., & Marx, B. D. (2010). Splines, knots, and penalties. Wiley Interdisciplinary Reviews: Computational Statistics, 2 (6), 637-653.
[23] Eilers, P. H., Marx, B. D., & Durbán, M. (2015). Twenty years of P-splines. Statistics and Operations Research Transactions, 39 (2), 0149-186.
[24] Marx, B. D., & Eilers, P. H. (1999). Generalized linear regression on sampled signals and curves: a P-spline approach. Technometrics, 41 (1), 1-13.
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    Sayed Meshaal El-sayed, Mohamed Reda Abonazel, Mohamed Metwally Seliem. (2020). B-spline Speckman Estimator of Partially Linear Model. International Journal of Systems Science and Applied Mathematics, 4(4), 53-59. https://doi.org/10.11648/j.ijssam.20190404.12

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    ACS Style

    Sayed Meshaal El-sayed; Mohamed Reda Abonazel; Mohamed Metwally Seliem. B-spline Speckman Estimator of Partially Linear Model. Int. J. Syst. Sci. Appl. Math. 2020, 4(4), 53-59. doi: 10.11648/j.ijssam.20190404.12

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    AMA Style

    Sayed Meshaal El-sayed, Mohamed Reda Abonazel, Mohamed Metwally Seliem. B-spline Speckman Estimator of Partially Linear Model. Int J Syst Sci Appl Math. 2020;4(4):53-59. doi: 10.11648/j.ijssam.20190404.12

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  • @article{10.11648/j.ijssam.20190404.12,
      author = {Sayed Meshaal El-sayed and Mohamed Reda Abonazel and Mohamed Metwally Seliem},
      title = {B-spline Speckman Estimator of Partially Linear Model},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {4},
      number = {4},
      pages = {53-59},
      doi = {10.11648/j.ijssam.20190404.12},
      url = {https://doi.org/10.11648/j.ijssam.20190404.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20190404.12},
      abstract = {The partially linear model (PLM) is one of semiparametric regression models; since it has both parametric (more than one) and nonparametric (only one) components in the same model, so this model is more flexible than the linear regression models containing only parametric components. In the literature, there are several estimators are proposed for this model; where the main difference between these estimators is the estimation method used to estimate the nonparametric component, since the parametric component is estimated by least squares method mostly. The Speckman estimator is one of the commonly used for estimating the parameters of the PLM, this estimator based on kernel smoothing approach to estimate nonparametric component in the model. According to the papers in nonparametric regression, in general, the spline smoothing approach is more efficient than kernel smoothing approach. Therefore, we suggested, in this paper, using the basis spline (B-spline) smoothing approach to estimate nonparametric component in the model instead of the kernel smoothing approach. To study the performance of the new estimator and compare it with other estimators, we conducted a Monte Carlo simulation study. The results of our simulation study confirmed that the proposed estimator was the best, because it has the lowest mean squared error.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - B-spline Speckman Estimator of Partially Linear Model
    AU  - Sayed Meshaal El-sayed
    AU  - Mohamed Reda Abonazel
    AU  - Mohamed Metwally Seliem
    Y1  - 2020/02/03
    PY  - 2020
    N1  - https://doi.org/10.11648/j.ijssam.20190404.12
    DO  - 10.11648/j.ijssam.20190404.12
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 53
    EP  - 59
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20190404.12
    AB  - The partially linear model (PLM) is one of semiparametric regression models; since it has both parametric (more than one) and nonparametric (only one) components in the same model, so this model is more flexible than the linear regression models containing only parametric components. In the literature, there are several estimators are proposed for this model; where the main difference between these estimators is the estimation method used to estimate the nonparametric component, since the parametric component is estimated by least squares method mostly. The Speckman estimator is one of the commonly used for estimating the parameters of the PLM, this estimator based on kernel smoothing approach to estimate nonparametric component in the model. According to the papers in nonparametric regression, in general, the spline smoothing approach is more efficient than kernel smoothing approach. Therefore, we suggested, in this paper, using the basis spline (B-spline) smoothing approach to estimate nonparametric component in the model instead of the kernel smoothing approach. To study the performance of the new estimator and compare it with other estimators, we conducted a Monte Carlo simulation study. The results of our simulation study confirmed that the proposed estimator was the best, because it has the lowest mean squared error.
    VL  - 4
    IS  - 4
    ER  - 

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Author Information
  • Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt

  • Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt

  • Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt

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