New Robust Estimators for the Nonparametric Regression Model: Application and Simulation Study

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-163
Published: Jul 16, 2025
Cite as:
Faten Mohamed Ali Soliman, Amany Moussa Mohamed, Mohamed R. Abonazel, New Robust Estimators for the Nonparametric Regression Model: Application and Simulation Study, Int. J. Anal. Appl., 23 (2025), 163.

Main Article Content

Faten Mohamed Ali Soliman, Amany Moussa Mohamed, Mohamed R. Abonazel

Abstract

This paper introduces new two robust kernel-based estimators (S Kernel and MM Kernel) for the nonparametric regression mode in the presence of outliers. Through comprehensive simulations, we evaluate their performance using Mean Squared Error (MSE), Mean Absolute Error (MAE), and Relative Efficiency (RE) under varying sample sizes and outlier contamination levels. Results demonstrate that robust estimators consistently outperform traditional kernel estimator, delivering the lowest estimation errors and highest efficiency, particularly in high-contamination scenarios. In contrast, the traditional kernel estimator proves highly sensitive to outliers. Also, our results highlight the superiority of the robust M Kernel estimator. This paper advances the field of robust nonparametric regression, offering practical solutions for datasets prone to outliers.

Article Details

References

  1. É.A. Nadaraya, On Non-Parametric Estimates of Density Functions and Regression Curves. Theory Probab. Appl. 1965, 10 (1), 186–190. https://doi.org/10.1137/1110024.
  2. G. Watson, Smooth Regression Analysis, Sankhyā Ser. A 26 (1964), 359-372.
  3. T. Gasser, H.G. Müller, Estimating Regression Functions and Their Derivatives by the Kernel Method, Scand. J. Stat. (1984), 171-185. https://www.jstor.org/stable/4615954.
  4. P. Hall, R.C.L. Wolff, Q. Yao, Methods for Estimating a Conditional Distribution Function, J. Amer. Stat. Assoc. 94 (1999), 154–163. https://doi.org/10.1080/01621459.1999.10473832.
  5. J. Fan, I. Gijbels, Local Polynomial Modelling and Its Applications, CRC Press, Boca Raton, 1996.
  6. S. Meshaal El-sayed, M. Reda Abonazel, M. Metwally Seliem, B-spline Speckman Estimator of Partially Linear Model, Int. J. Syst. Sci. Appl. Math. 4 (2019), 53-59. https://doi.org/10.11648/j.ijssam.20190404.12.
  7. M.R. Abonazel, N. Helmy, A. Azazy, The Performance of Speckman Estimation for Partially Linear Model Using Kernel and Spline Smoothing Approaches, Int. J. Math. Arch. 10 (2019), 10-18.
  8. M. Abonazel, A. Rabie, The Impact of Using Robust Estimations in Regression Models: An Application on the Egyptian Economy, J. Adv. Res. Appl. Math. Stat. 4 (2019), 8-16.
  9. M. M. Elgohary, M. R. Abonazel, N. M. Helmy, A. R. Azazy, New Robust-Ridge Estimators for Partially Linear Model, Int. J. Appl. Math. Res. 8 (2019), 46-52. https://doi.org/10.14419/ijamr.v8i2.29932.
  10. W.S. Cleveland, Robust Locally Weighted Regression and Smoothing Scatterplots, J. Am. Stat. Assoc. 74 (1979), 829-836. https://doi.org/10.1080/01621459.1979.10481038.
  11. B.W. Silverman, Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting, J. R. Stat. Soc. Ser. B: Stat. Methodol. 47 (1985), 1-21. https://doi.org/10.1111/j.2517-6161.1985.tb01327.x.
  12. J. Fan, P. Hall, On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications, Ann. Stat. 22 (1994), 867-885. https://doi.org/10.1214/aos/1176325499.
  13. J.W. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading, 1977.
  14. I.R. Ghement, M. Ruiz, R. Zamar, Robust Estimation of Error Scale in Nonparametric Regression Models, J. Stat. Plan. Inference 138 (2008), 3200-3216. https://doi.org/10.1016/j.jspi.2008.01.005.
  15. H.F.F. Mahmoud, B. Kim, I. Kim, Robust Nonparametric Derivative Estimator, Commun. Stat. - Simul. Comput. 51 (2020), 3809-3829. https://doi.org/10.1080/03610918.2020.1722836.
  16. G. Wang, F. Manhardt, F. Tombari, X. Ji, GDR-Net: Geometry-Guided Direct Regression Network for Monocular 6D Object Pose Estimation, in: 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, Nashville, TN, USA, 2021: pp. 16606–16616. https://doi.org/10.1109/CVPR46437.2021.01634.
  17. B. Zhu, J. Jiao, M.I. Jordan, Robust Estimation for Non-Parametric Families via Generative Adversarial Networks, in: 2022 IEEE International Symposium on Information Theory (ISIT), IEEE, Espoo, Finland, 2022: pp. 1100–1105. https://doi.org/10.1109/ISIT50566.2022.9834844.
  18. M. Salibian-Barrera, Robust Nonparametric Regression: Review and Practical Considerations, Econ. Stat. 4 (2023), 4. https://doi.org/10.1016/j.ecosta.2023.04.004.
  19. M.R. Abonazel, A Practical Guide for Creating Monte Carlo Simulation Studies Using R, Int. J. Math. Comput. Sci. 4 (2018), 18-33.
  20. A. Kamel, M.R. Abonazel, A Simple Introduction to Regression Modeling Using R, Comput. J. Math. Stat. Sci. 2 (2023), 52-79. https://doi.org/10.21608/cjmss.2023.189834.1002.
  21. M.D. Cattaneo, M. Jansson, X. Ma, Lpdensity: Local Polynomial Density Estimation and Inference, arXiv:1906.06529 (2019). http://arxiv.org/abs/1906.06529.
  22. S. Calonico, M.D. Cattaneo, M.H. Farrell, Nprobust: Nonparametric Kernel-Based Estimation and Robust Bias-Corrected Inference, arXiv:1906.00198 (2019). http://arxiv.org/abs/1906.00198.
  23. P. Cortez, A. Cerdeira, F. Almeida, T. Matos, J. Reis, Modeling Wine Preferences by Data Mining from Physicochemical Properties, Decis. Support Syst. 47 (2009), 547-553. https://doi.org/10.1016/j.dss.2009.05.016.
  24. M.R. Abonazel, A.H. Youssef, E.G. Ahmed, On Robust M, S and Mm Estimations for the Poisson Fixed Effects Panel Model with Outliers: Simulation and Applications, J. Stat. Comput. Simul. 95 (2025), 1190-1207. https://doi.org/10.1080/00949655.2024.2449102.
  25. M.M. Seliem, S.M. El-Sayed, M.R. Abonazel, Developing a Semi-Parametric Zero-Inflated Beta Regression Model Using P-Splines: Simulation and Application, Stat. Optim. Inf. Comput. 13 (2024), 1103-1119. https://doi.org/10.19139/soic-2310-5070-2220.