The Nonparametric Kernel Method using Nadaraya-Watson, Priestley-Chao and Gasser-Muller Estimators for the Estimation of the Rainfall Data in Lampung

International Journal of Mathematics Trends and Technology (IJMTT)
© 2022 by IJMTT Journal
Volume-68 Issue-8
Year of Publication : 2022
Authors : Herawati. N, Sayuti. S.F, Nisa. K, Setiawan. E

How to Cite?

Herawati. N, Sayuti. S.F, Nisa. K, Setiawan. E, "The Nonparametric Kernel Method using Nadaraya-Watson, Priestley-Chao and Gasser-Muller Estimators for the Estimation of the Rainfall Data in Lampung," International Journal of Mathematics Trends and Technology, vol. 68, no. 8, pp. 12-20, 2022. Crossref,

There are two methods used in regression analysis, namely parametric and nonparametric. Parametric regression, assumes many assumptions, such as the normality of data, the distribution of data that forms a certain pattern, and others. In nonparametric regression, these assumptions do not have to be fulfilled. To estimate the nonparametric regression function, smoothing techniques are needed, one of which is the kernel method. In this study, the nonparametric regression estimators of Nadaraya-Watson, Priestley & Chao and Gasser & Müller were compared based on the smallest GCV value and the best kernel estimator was determined by the smallest error rate in the rainfall data in the province of Lampung. The results showed that the Nadaraya-Watson estimator was determined as the best estimator with a bandwidth (h) of 5.564 and had the smallest error value compared to the other two methods, namely MSE=5.9773 and MAPE=0.1935.

Keywords : Kernel estimator, Bandwidth, Nadaraya-Watson, Priestley-Chao, Gasser-Muller, MSE, MAPE.


[1] S. J. Sheather, Density estimation, “Statistical Science,” vol.19,no.4, pp. 588–597, 2004.
[2] S. Halim, and I. Bisono, “Kernel functions on nonparametric regression methods and their applications,” Journal of Industrial Engineering, vol. 8, no.1, pp. 73–81, 2006.
[3] N. Herawati, K. Nisa, and E. Setiawan, “The Optimal Bandwidth for Kernel Density Estimation of Skewed Distribution: A Case Study on Survival Time Data of Cancer Patients,” Proc. National Seminar on Quantitative Method, vol.1, no.1, pp. 380-388, 2017.
[4] M.A. Dakhil and J. N. Hussain, “A Comparative Study of Nonparametric Kernel estimators with Gaussian Weight Function,” Journal of Physics: Conference Series, vol.18, no.1, pp. 1-10, 2021.
[5] I. Puspitasari, Suparti, and Y. Wilandari, “Analysis of the Composite Stock Price Index (JCI) Using the Kernel Regression Model,” Gaussian Journal, vol.1, no.1, pp. 93–102, 2012.
[6] N.A.K.. Rifai, Kernel Nonparametric Regression Approach on Composite Stock Price Index Data, STATISTICS,” Journal of Theoretical Statistics and Its Applications, vol.19, no.1, pp. 53–61, 2019.
[7] D. Suparti, I. P. Safitri, Sari, and A. R. Devi, “Analysis of Inflation Data in Indonesia Using the Kernel Regression Model,” Proceedings of the National Seminar on Statistics, pp. 499-509, 2013.
[8] Suparti, “Analysis of Inflation Data in Indonesia After the 2013 Increase in TDL and BBM Using the Kernel Regression Model,” Statistical Media, vol.6, no.2, pp.103-112, 2013
[9] G. Rubio, H. Pomares, L. J. Herrera, and I. Rojas, “Kernel methods applied to time series forecasting,” Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics, Springer-Verlag, pp.782–789, 2007
[10] S. Saidi, N. Herawati, K. Nisa, and E. Setiawan, “Nonparametric Modeling Using Kernel Method for the Estimation of the Covid-19 Data in Indonesia During 2020,” International Journal of Mathematics Trends and Technology, vol. 67, no.6, pp. 136-144, 2021.
[11] W. Syisliawati, Wibowo, and I. N. Budiantara, “Regression Spline Truncated Curve in Nonparametric Regression,” Proc. of 3rd International Conference on Research Implementation and Education of Mathematics and Science, vol.18, no.1 , pp. 115-122, 2016.
[12] M. Y. Mustafa and Z. Y. Algamal, “Smoothing Parameter Selection in Kernel Nonparametric Regression Using Bat Optimization Algorithm,” Journal of Physics: Conference Series, vol.1897, no.1,pp. 1-9, 1897.
[13] S. Molchanov and S. N. Chiu, “Smoothing Techniques and Estimation Methods for Nonstationary Boolean Models with Applications to Coverage Processes,” Biometrics, vol.87, no.2, pp. 265-283, 2012.
[14] M. Rosenblatt, “Remarks on Some Nonparametric Estimates of a Density Function,” The Annals of Mathematical Statistics, vol.33 , pp.832-837, 1956
[15] E. Parzen, “On Estimation of a Probability Density Function and Mode,” The Annals of Mathematical Statistics, vol.33 , pp.1065-1076, 1962.
[16] W.R. Schucany, “On nonparametric regression with higher-order kernels,” Journal of Statistical Planning and Inference, vol.23, no.2 , pp. 145-151, 1989.
[17] S. Weglarczyk, “Kernel Density Estimation and Its Application,” ITM Web of Conferences, vol.23, no.374 , pp. 1-8, 2018.
[18] Nadaraya, E. A, “Some new estimates for distribution functions,” Theory of Probability & Its Applications, vol.9, no.3, pp. 497–500, 1964.
[19] S. Demir and O. Toktamis, On The Adaptive Nadaraya-Watson Kernel Regression Estimators, Hacettepe Journal of Mathematics and Statistics, vol.39, no.3, pp. 429-437, 2010.
[20] K. Konecna, The Priestley-Chao Estimator of Conditional Density with Uniformly Distributed Random Design, statistics, vol.98, no.3, pp. 283-294, 2019.
[21] M. E. Priestley, and M. T. Chao, “Nonparametric function Fitting, Journal of The Royal Statistical Society, vol.34 ,pp. 385-392, 1972.
[22] T. Gasser, and H.-G. Müller,” Kernel Estimation of Regression Functions. In Smoothing Techniques for Curve Estimation, Springer, Berlin, pp.23-68, 1979.
[23] P. Babilua, E. Nadaraya, and G. Sokhadze, “Functionals of Gasser–Muller Estimators,” Turkish Journal of Mathematics, vol.38, no.1, pp. 1090–1101, 2014.
[24] M. C. Jones, J. S. Marron, and S. J. Sheather, “A Brief Survey of Bandwidth Selection for Density Estimation,” Journal of the American Statistical Association, vol.91, no.433, pp. 401-407, 1996.
[25] L. R. Cheruiyot, G. O. Orwa, and O. E. Otieno, “Kernel Function and Nonparametric Regression Estimation: Which Function Is Appropriate,” African Journal of Mathematics and Statistics Studies, vol.3, no.3, pp. 51-59, 2020.
[26] O.M. Eidous, M.A.A.S Maie, and M.A Ebrahem, “A Comparative Study for Bandwidth Selection in Kernel Density Estimation,” Journal of Modern Applied Statistical Methods: JMASM, vol.9, no.1, pp.263-273, 2015.
[27] H. Dhaker, E. H. Deme, P. Ngom, and M. Mbodj, “New Approach for Bandwidth Selection in The Kernel Density Estimation Based on Β-Divergence,” Journal of Mathematical Sciences: Advances and Applications, vol.51, no.1, pp. 57–83, 2018.
[28] G. Kauermann and J. D. Opsomer, Generalized Cross-Validation for Bandwidth Selection of Backfitting Estimates in Generalized Additive Models, Journal of Computational and Graphical Statistics, vol.13, no.1, pp. 66-89, 2014.