Parametric solution of p-norm semiparametric regression model


In this paper, using the kernel weight function, we obtain the parameter estimation of p-norm distribution in semi-parametric regression model, which is effective to decide the distribution of random errors. Under the assumption that the distribution of observations is unimodal and symmetry, this method can give the estimates of X, S and σ. Finally, three experiment are constructed to explain this method. When there is model error, the traditional least squares method is estimated to be distorted. In the first experiment, the parameter has a true value of 1, and the least squares method and the proposed new method are estimated to be 0.4883 and 0.9898 respectively. The proposed method is applicable to the different distribution of random errors. In the example of this article, it has also been reflected.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Aerts M, Claeskens G, Wand MP (2002) Some theory for penalized spline generalized additive models [J]. J Stat Plan Infer 103(1–2):455–470

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cui X, Yu Z, Tao B, et al. (2001) Generalized surveying adjustment [J]. Wuhan Technique University of Surveying and Mapping, Wuhan,: 103–104

  3. 3.

    Eubank RL, Kambour EL, Kim JT et al (1998) Estimation in partially linear models [J]. Comput Stat Data Anal 29(1):27–34

    MathSciNet  Article  Google Scholar 

  4. 4.

    Fischer B, Hegland M (1999) Collocation, filtering and nonparametric regression. PartI, ZfV, 17-24

  5. 5.

    Green PJ, Silverman BW (1993) Nonparametric regression and generalized linear models: a roughness penalty approach. CRC Press

  6. 6.

    Hong SY (2002) Normal approximation rate and bias reduction for data-driven kernel smoothing estimator in a semiparametric regression model. J Multivar Anal 80.1:1–20

    MathSciNet  Article  Google Scholar 

  7. 7.

    Müller M (2000) Semiparametric extensions to generalized linear models [J]. Kumulative Habilitationsschrift, Wirtschaftswissenschaftliche Fakultät, Humboldt Universität zu Berlin, Berlin

    Google Scholar 

  8. 8.

    Nelder JA, Wedderburn RWM (1972) Generalized linear models [J]. J R Stat Soc 135(3):370–384

    Google Scholar 

  9. 9.

    Pan X, Sun H (2004a) Statistical property of semiparametric estimators with positively definite regular matrix. ACTA Geodaetica & Cartographica Sinica 33:228–233

    Google Scholar 

  10. 10.

    Pan X, Sun H (2004b) Two stage estimation of parameter for semiparametric survey. Adjustment Models, Science of Sueveying and Mapping 29:19–22

    Google Scholar 

  11. 11.

    Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge series in statistical and probabilistic mathematics 12. Cambridge Univ. Press. Mathematical Reviews (MathSciNet): MR1998720, Cambridge

    Google Scholar 

  12. 12.

    Schimek MG (2000) Estimation and inference in partially linear models with smoothing splines [J]. J Stat Plan Infer 91(2):525–540

    MathSciNet  Article  Google Scholar 

  13. 13.

    Shi P, Tsai C-L (1999) Semiparametric regression model selections. J Stat Plan Infer 77.1:119–139

    MathSciNet  Article  Google Scholar 

  14. 14.

    Sugiyama M, Ogawa H (2002) A unified method for optimizing linear image restoration filters [J]. Signal Process 82(11):1773–1787

    Article  Google Scholar 

  15. 15.

    Sun HY (1994) Theory of p-norm distribution and application of surveying data processing [J]. WTUSM Press 2:172–174

    Google Scholar 

  16. 16.

    Sun H, Yun WU (2002) Semiparametric regression and model refining. Geo-Spatial Inform Sci 5(4):10–13

    Article  Google Scholar 

  17. 17.

    Wang Q, Rao JNK (2002) Empirical likelihood-based inference in linear errors-in-covariables models with validation data [J]. Biometrika 89(2):345–358

    MathSciNet  Article  Google Scholar 

  18. 18.

    Yu Y, Ruppert D (2002) Penalized spline estimation for partially linear single-index models [J]. J Am Stat Assoc 97(460):1042–1054

    MathSciNet  Article  Google Scholar 

  19. 19.

    Zhao XM, Wei XY (2003) Asymptotic efficiency of L-estimators in a semi-parametric regression model [J]. J Syst Sci Math Sci (1): 136–144

Download references


This study was funded by the National Natural Science Foundation of China (41,476,087, 4187 4009).

Author information



Corresponding author

Correspondence to Haijun Huang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xu, J., Huang, H. & Pan, X. Parametric solution of p-norm semiparametric regression model. Multimed Tools Appl 78, 30127–30139 (2019).

Download citation


  • p-norm distributions
  • Semi-parametric regression
  • Kernel weight function
  • Maximum likelihood adjustment