Optimizing bandwidth parameter estimation for non-parametric regression using fixed-form threshold with Dmey and Coiflet wavelets

Abstract

The article aims to reduce the effect of data noise or outliers and estimate the optimal bandwidth parameter used in nonparametric regression models using a proposed method based on wavelet analysis, specifically Dmey and Coiflet wavelets with fixed-form threshold and apply the soft threshold, particularly when the data have long-tailed and multimodal distributions (abnormal distribution). The fixed-form threshold level value estimates the bandwidth instead of the classical method (geometric, arithmetic mean, range, and median). A simulation study was used to examine the suggested method, comparing it with four other Nadaraya-Watson kernel estimators (classical techniques), using a MATLAB language created especially for this purpose with actual data. The findings show that the suggested method outperforms classical methods for all cases of simulations and real data in accurately estimating the bandwidth parameter of the non-parametric regression kernel function based on the mean square error criterion.

Keywords

• Non-parametric regression
• kernel estimator
• bandwidth parameter
• fixed-form threshold
• wavelets

References

1. [1] I. Abramson, On bandwidth variation in kernel estimates—a square root law, Ann. Stat. 10:1217–1223, 1982.
2. [2] T. H. Ali, Using proposed nonparametric regression models for clustered data (a simulation study), ZANCO J. Pure Appl. Sci. 29:78–87, 2017.
3. [3] T. H. Ali, Modification of the adaptive Nadaraya-Watson kernel method for nonparametric regression (simulation study), Commun. Stat. Simul. Comput. 51:391–403, 2022.
4. [4] T. H. Ali, H. A. A.-M. Hayawi, and D. Shaker Botani, Estimation of the bandwidth parameter in Nadaraya-Watson kernel non-parametric regression based on universal threshold level, Commun. Stat. Simul. Comput. 52:1476–1489, 2023.
5. [5] T. H. Ali and J. R. Qadir, Using wavelet shrinkage in the Cox proportional hazards regression model (simulation study), Iraq J. Stat. Sci. 19:17–29, 2022.
6. [6] T. H. Ali and D. M. Saleh, Comparison between wavelet Bayesian and Bayesian estimators to remedy contamination in linear regression model, PalArch J. Egypt. Egyptol. 18, 2021.
7. [7] K. H. Aljuhani and L. I. A. Turk, Modification of the adaptive Nadaraya-Watson kernel regression estimator, Sci. Res. Essays 9:966–971, 2014.
8. [8] I. L. Cascio, Wavelet analysis and denoising: New tools for economists, 2007.

9. [9] A. Christmann and I. Steinwart, Consistency and robustness of kernel-based regression in convex risk minimization, Bernoulli 13:799–819, 2007.
10. [10] R.R. Coifman and M.V. Wickerhauser, Entropy-based algorithms for best basis selection, IEEE Trans. Inf. Theory 38(2):713–718, 1992.
11. [11] S. Demir and Toktamı􀀀ş, On the adaptive Nadaraya-Watson kernel regression estimators, Hacet. J. Math. Stat. 39:429–437, 2010.
12. [12] R. Eubank, Spline Smoothing and Nonparametric Regression, Dekker, 1988.
13. [13] J. H. Friedman and W. Stuetzle, Projection pursuit regression, J. Am. Stat. Assoc. 76:817–823, 1981.
14. [14] Y. A. Hassan and M. Y. Hmood, Estimation of return stock rate by using wavelet and kernel smoothers, Period. Eng. Nat. Sci. 8(2):602–612, 2020.
15. [15] W. Härdle and G. Kelly, Non-parametric kernel regression estimation optimal choice of bandwidth, Stat. 18:21–35, 1987.
16. [16] D. Li and R. Li, Local composite quantile regression smoothing for Harris recurrent Markov processes, J. Econom. 194:44–56, 2016.
17. [17] H. Läuter, Silverman, B. W.: Density estimation for statistics and data analysis. Chapman & Hall, London – New York, 1986, Biometr. J. 30:876–877, 1988.
18. [18] M. Y. Mustafa and Z. Y. Algamal, Smoothing parameter selection in kernel nonparametric regression using bat optimization algorithm, J. Phys.: Conf. Ser. 1897, 2021.
19. [19] E. A. Nadaraya, On estimating regression, Theory Probab. Appl. 9:141–142, 1964.
20. [20] D. H. Rashid, M. Y. Hmood, and S. K. Hamza, Nadaraya-Watson estimator a smoothing technique for estimating regression function, J. Econ. Adm. Sci. 18(65):283, 2012.
21. [21] D. W. Scott and G. R. Terrell, Biased and unbiased cross-validation in density estimation, J. Am. Stat. Assoc. 82:1131–1146, 1987.
22. [22] S. Shahzadi, U. Shahzad, and N. Koyuncu, On the adaptive Nadaraya-Watson kernel estimator for the discontinuity in the presence of jump size, SDU J. Nat. Appl. Sci. 22:511–520, 2018.
23. [23] M. P. Wand and M. C. Jones, Kernel Smoothing, Chapman and Hall, 1995.
24. [24] G. S. Watson, Smooth regression analysis, Sankhyā Ser. A 26:359–372, 1964.
25. [25] S. Weisberg, Applied Linear Regression, Wiley, New York, 1985.