ASSESSMENT OF BANDWIDTH SELECTION STRATEGIES TO SMOOTH THE MEAN REGRESSION FUNCTION

Tarek A. Elghazali; Fatima H. Alhamroush; Fawzia A. Altobly; Ahmed M. Mami

doi:10.64943/ljacs.2026.010213

Authors

Tarek A. Elghazali Department of Statistics, Faculty of Science, University of Benghazi, Benghazi, Libya https://orcid.org/0000-0001-9461-5644
Fatima H. Alhamroush Department of Statistics, Faculty of Science, University of Benghazi, Benghazi, Libya https://orcid.org/0009-0005-1882-0634
Fawzia A. Altobly Department of Statistics, Faculty of Science, University of Benghazi, Benghazi, Libya https://orcid.org/0009-0002-1207-5125
Ahmed M. Mami Department of Statistics, Faculty of Science, University of Benghazi, Benghazi, Libya https://orcid.org/0009-0007-6246-212X

DOI:

https://doi.org/10.64943/ljacs.2026.010213

Keywords:

Local Polynomial Kernel Estimators, Direct Plug-in Selection Strategy, Residual Square Criterion, the Cross-validation criterion, the Rule of Thumb criterion.

Abstract

Non-parametric regression is a type of regression analysis in which the functional form of the relationship between the response variable, and the associated predictor variable does not to be specified in order to fit a model to a set of data. There are many different methods for non-parametric regression. In this paper, we have used the local polynomial kernel estimator with the optimal choice of the smoothing parameter(s). Choosing the optimal smoothing parameter(s), which is usually called the bandwidth(s), was considered to be one of the most important issues when using the kernel-based estimators. Moreover, the optimal bandwidth(s) played a great role to uncover the structure of the underlying data. Here, we used simulation approach to make comparisons between four different strategies for selecting the optimal bandwidth(s), namely the Direct Plug In bandwidth selection, the Residual Square Criterion, the Cross-validation criterion, and the Rule of Thumb criterion. Within the context of these four strategies of selecting the optimal bandwidth(s), there are two different design settings: Fixed Design and Random Design. In addition, four different regression functions have been tested in order to smooth the mean regression function and their first and second derivatives. We have investigated the effect of three important elements: The size of sample ( ); the distribution of the errors' (normal or non-normal distribution); and the kind of design (fixed and random). Having conducted simulation studies, interesting results have been achieved.

Keywords: Local Polynomial Kernel Estimators, Direct Plug-in Selection Strategy, Residual Square Criterion, the Cross-validation criterion, the Rule of Thumb criterion.

References

1. Gasser, T., & Müller, H.-G. (1984). Estimating regression functions and their derivatives by the kernel method. Scandinavian Journal of Statistics, 171–185.

2. Hastie, T., Tibshirani, R., Friedman, J. H., & Friedman, J. H. (2017). The Elements of Statistical Learning Data Mining, Inference, and Prediction (2 (ed.)). Springer.

3. James, G., Witten, D., Hastie, T., & T. (2023). An introduction to statistical learning with applications in R (2nd ed.). Springer.

4. Härdle, W. (1990). Applied nonparametric regression (Issue 19). Cambridge university press.

5. Fan, J. (2018). Local polynomial modelling and its applications: monographs on statistics and applied probability 66. Routledge. https://doi.org/10.1201/9780203748725

6. Yousuf, A., Jamal Saadoon , A., & Said Khan , F. (2026). Phytochemical Composition and Heavy Metals Analysis of Crude Ethanolic Extract of Geranium wallichianum, Elaeagnus parvifolia, and Taraxacum officinale. Libyan Journal of Applied and Contemporary Sciences, 1(1), 30–37. https://doi.org/10.64943/ljacs.2026.010104

7. Mami, A. M. (2002). Local polynomial regression with applications to both independent and longitudinal data. University of Manchester.

8. Chacón, J. E., & Duong, T. (2018). Multivariate kernel smoothing and its applications. Chapman and Hall/CRC.

9. Guidoum, A. C. (2015). Kernel estimator and bandwidth selection for density and its derivatives. Department of Probabilities and Statistics, University of Science and Technology, Houari Boumediene, Algeria.

10. Sheather, S. J., & Jones, M. C. (1991). A reliable data‐based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society: Series B (Methodological), 53(3), 683–690.

11. Liu, S., & Charnigo, R. (2023). Tuning parameter selection for nonparametric derivative estimation in random design. Statistics, 57(6), 1402–1425. https://doi.org/10.1080/02331888.2023.2278042

12. Loader, C. R. (1999). BANDWIDTH SELECTION: CLASSICAL OR PLUG-IN? The Annals of Statistics, 27(2), 415–438.

13. Ruppert, D., Sheather, S. J., & Wand, M. P. (1995). An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association, 90(432), 1257–1270.

14. Henderson, D. J., & Parmeter, C. F. (2015). Applied nonparametric econometrics. Cambridge University Press.

15. Li, Q., & Racine, J. S. (2007). Nonparametric econometrics: theory and practice. Princeton University Press.

16. 16. Mallows, C. L. (1995). More comments on Cp. Technometrics, 37(4), 362–372.

17. Park, B. U., & Marron, J. S. (1990). Comparison of data-driven bandwidth selectors. Journal of the American Statistical Association, 85(409), 66–72.

18. Hall, P., & Marron, J. S. (1990). On variance estimation in nonparametric regression. Biometrika, 415–419.

19. Zhou, J., & Parmeter, C. F. (2025). Data-driven resistant kernel regression. Journal of Nonparametric Statistics, 37(1), 33–59. 10.1080/ https://doi.org/10485252.2024.2335494

20. Hossen Mohammed Al-Mahdi Al-Sharif. (2025). Analysis and Evaluation of ARIMA and SARIMA Models Performance in Time Series Forecasting: An Applied Study. Libyan Journal of Medical and Applied Sciences, 3(2), 146–157. https://doi.org/10.64943/ljmas.v3i2.89

21. Fowzi S. Alarabi. (2025). Investigation of Scattered Power and Cross-Polarization Effects on Linearly Polarized Waves under Sandstorm Conditions. Libyan Journal of Medical and Applied Sciences, 3(3), 81–90. https://doi.org/10.64943/ljmas.v3i3.140

22. García-Portugués, E., & Prieto-Alaiz, M. (2022).A review and comparison of bandwidth selection criteria for kernel regression and local polynomial estimators. Statistical Science, 37(4), 582–601. https://doi.org/10.1214/22-STS856

23. Abdelsalam Milad, F. (2026). Optimization of Laser-Induced Optical Response in Advanced Materials for High-Performance Photonic Applications. Libyan Journal of Applied and Contemporary Sciences, 1(2), 70–89. https://doi.org/10.64943/ljacs.2026.010207

24. Zhang, Y., & Chen, X. (2022). A comparative study of bandwidth selection criteria in kernel regression under fixed and random designs. Communications in Statistics - Simulation and Computation, 51(8), 4520–4539. https://doi.org/10.1080/03610918.2020.1741623

25. Kong, D., Zhao, S., & Wang, W. (2024). Nonparametric derivative estimation with bimodal kernels under correlated errors: D. Kong et al. Computational Statistics, 39(4), 1847–1865. https://doi.org/10.1007/s00180-023-01419-4

26. Sisheng, L., & Jing, Y. (2024). Kernel regression for estimating regression function and its derivatives with unknown error correlations: L. Sisheng, Y. Jin. Metrika, 87(1), 1–20. https://doi.org/10.1007/s00180-023-01419-4

27. Najeeb Ahmed , O., & Amer Mohammed, A. (2026). Intelligent Deep Learning-based Systems to Detect Early Cyber Attacks in IoT Networks. Libyan Journal of Applied and Contemporary Sciences, 1(1), 38–52. https://doi.org/10.64943/ljacs.2026.010105

28. Barreiro-Ures, D., Cao, R., & Francisco-Fernández, M. (2023).Subsampling-based cross-validation bandwidth selection in non-parametric kernel regression under fixed and random designs. Computational Statistics & Data Analysis, 181, 107689. https://doi.org/10.1016/j.csda.2022.107689

29. Kassi, O., & Wang, S. G. W. (2024).Structural adaptation and rate-accelerated bandwidth selection in non-parametric local polynomial smoothing. Journal of Nonparametric Statistics, 36(3), 512–535. https://doi.org/10.1080/10485252.2024.2315891