Boundary Estimation with the Fuzzy Set Regression Estimator
Keywords:
Fuzzy set regression estimator, boundary estimation
Abstract
In order to extend the properties of the fuzzy set regression estimation method and provide new results related to the nonparametric regression estimation problems not based on kernels, this paper analyzes the possible boundary effects, if any, of the fuzzy set regression estimator and presents a criterion to remove it. Moreover, a boundary fuzzy set estimator is proposed which is defined as a particular class of fuzzy set regression estimators, where the bias, variance, mean squared error and function that minimizes the mean squared error of the proposed estimator are given. Finally, these theoretical findings are illustrated through some numerical examples, and with one real data example. Simulations show that the proposed estimator has better performance at points near zero in a spread neighborhood of the smoothing parameter, when it is compared with a general boundary kernel regression estimator for the two regression models and two density functions considered. The previously exposed represents the natural extension of the recent results to the boundary fuzzy set density estimator case.
References
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http://www3.stat.sinica.edu.tw/statistica/j6n4/j6n414/j6n414.htm
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[22] I. Salgado-Ugarte, M. Shimizu and T. Taniuchi (1996). Nonparametric regression: Kernel, WARP, and k-NN estimators, S.T.B., 5. https://EconPapers.repec.org/Re-PEc:tsj:stbull:y:1996:v:5:i:30:snp10
[23] G. Schmidt, R. Mattern and F. Schüler (1981). Biomechanical investigation to determine physical and traumatological differentiation criteria for the maximum load capacity of head and vertebral column with and without protective helmet under effects of impact. EEC Research Program on Biomechanics of Impac, Final Report Phase III, Project 65, Institut für Rechtsmedizin, Universität Heidelberg.
[24] K. Souraya, A. Sayah and Y. Djabrane (2015). General method of boundary correction in kernel regression estimation, Afr. Stat., 10, 739–750. DOI: 10.16929/as/2015.729.66
[25] I. Sriliana, I . N. Budiantara and V. Ratnasari (2022). A Truncated Spline and Local Linear Mixed Estimator in Nonparametric Regression for Longitudinal Data and Its Application, Symmetry, 14 (12), 2687. DOI: 10.3390/sym14122687
[26] C. J. Stone (1980). Optimal rates of convergence for nonparametric estimators, Ann. Statist., 8, 1348–1360. DOI: 10.1214/aos/1176345206
[27] G. S. Watson (1964). Smooth regression analysis, Sankhyā, Ser. A, 26, 359–372. https://www.jstor.org/stable/25049340
[28] L. A. Zadeh (1965). Fuzzy sets, Inform. Control, 8, 338–353.
[29] S. Zhang and R. J. Karunamuni (1998). On kernel density estimation near endpoints, J. Stat. Plan. Infer., 70 (2), 301–316. DOI: 10.1016/S0378-3758(97)00187-0
[30] S. Zhang, R. J. Karunamuni and M. C. Jones (1999). An Improved Estimator of the Density Function at the Boundary, J. Am. Stat. Assoc. 94 (448), 1231–1240. DOI: 10.1080/01621459.1999.10473876
[2] L. R. Cheruiyot (2020). Local Linear Regression Estimator on the Boundary Correction in Nonparametric Regression Estimation, JSTA, 19 (3), 460–471. DOI: 10.2991/jsta.d.201016.001
[3] L. R. Cheruiyot, O. R. Otieno and G. O. Orwa (2019). A Boundary Corrected Non-Parametric Regression Estimator for Finite Population Total,Int. J. Probab. Stat., 8 (3), 1–83. DOI: 10.5539/ijsp.v8n3p83
[4] F. Comte, and N. Marie (2021). On a Nadaraya-Watson estimator with two bandwidths, Electron. J. Statist, 15 (1), 2566–2607. DOI: 10.1214/21-EJS1849
[5] J. Fajardo (2012). A Criterion for the Fuzzy Set Estimation of the Regression Function, J. Prob. Stat., 2012. DOI: 10.1155/2012/593036
[6] J. Fajardo (2014). A criterion for the fuzzy set estimation of the density function, Braz. J. Prob. Stat., 28 (3), 301–312. DOI: 10.1214/12-BJPS208
[7] J. Fajardo (2018). Comparison of the Mean Square Errors of the Fuzzy Set Regression Estimator and Local Linear Regression Smoothers, Int. J. Math. Stat., 19 (1), 74–89. http://www.ceser.in/ceserp/index.php/ijms/article/view/5355
[8] J. Fajardo and P. Harmath (2021). Boundary estimation with the fuzzy set density estimator, METRON, 79, 285–302. DOI: 10.1007/s40300-021-00210-z
[9] J. Fajardo R. Rı́os and L. Rodrı́guez (2010). Properties of convergence of a fuzzy set estimator of the regression function, J. Stat., Adv. Theory Appl., 3 (2), 79–112. Zbl 05800441
[10] J. Fajardo, R. Rı́os and L. Rodrı́guez (2012). Properties of convergence of a fuzzy set estimator of the density function, Braz. J. Prob. Stat., 26 (2), 208–217. DOI: 10.1214/10-BJPS130
[11] M. Falk and F. Liese (1998). Lan of thinned empirical processes with an application to fuzzy set density estimation, Extremes, 1 (3), 323–349. DOI: 10.1023/A:1009981817526
[12] F. Ferraty, V. Núñez-Antón, and P. Vieu (2001). Regresión No Paramétrica: Desde la Dimensión uno Hasta la Dimensión Infinita, Servicio Editorial de la Universidad del Pais Vasco, Bilbao.
[13] P. Hall and B. U. Park (2002). New methods for bias correction at endpoints and boundaries, Ann. Statist., 30 (5), 1460–1479. DOI: 0.1214/aos/1035844983
[14] W. Härdle (1990). Applied nonparametric regression, Cambridge University Press.
[15] R. Hidayat, I. N. Budiantara, B. W. Otok and V. Ratnasari (2021). The regression curve estimation by using mixed smoothing spline and kernel (MsS-K) model, Commun. Stat. Theory Methods, 50 (17), 3942–3953. DOI: 10.1080/03610926.2019.1710201
[16] M. C. Jones and P. J. Foster (1996). A simple nonnegative boundary correction method for kernel density estimation, Stat. Sinica, 6 (4), 1005–1013.
http://www3.stat.sinica.edu.tw/statistica/j6n4/j6n414/j6n414.htm
[17] R. J. Karunamuni and T. Alberts (2005). On boundary correction in kernel density estimation, Stat. Methodol., 2 (3), 191–212. DOI: 10.1016/j.stamet.2005.04.001
[18] B. C. Kouassi, O. Hili and E. Katchekpele (2021). Nadaraya-Watson estimation of a non-parametric autoregressive mode, MJM, 9 (4), 251–258. DOI: 10.26637/mjm904/009
[19] Y. Linke, I. Borisov, P. Ruzankin, V. Kutsenko, E. Yarovaya and S. Shalnova (2022). Universal Local Linear Kernel Estimators in Nonparametric Regression, Mathematics, 10 (15), 2693. DOI: 10.3390/math10152693
[20] E. A. Nadaraya (1964). On estimating regression, Theory Probab. Appl., 9, 141–142. DOI: 10.1137/1109020
[21] R. D. Reiss (1993). A Course on Point Processes, Springer Series in Statistics, New York.
[22] I. Salgado-Ugarte, M. Shimizu and T. Taniuchi (1996). Nonparametric regression: Kernel, WARP, and k-NN estimators, S.T.B., 5. https://EconPapers.repec.org/Re-PEc:tsj:stbull:y:1996:v:5:i:30:snp10
[23] G. Schmidt, R. Mattern and F. Schüler (1981). Biomechanical investigation to determine physical and traumatological differentiation criteria for the maximum load capacity of head and vertebral column with and without protective helmet under effects of impact. EEC Research Program on Biomechanics of Impac, Final Report Phase III, Project 65, Institut für Rechtsmedizin, Universität Heidelberg.
[24] K. Souraya, A. Sayah and Y. Djabrane (2015). General method of boundary correction in kernel regression estimation, Afr. Stat., 10, 739–750. DOI: 10.16929/as/2015.729.66
[25] I. Sriliana, I . N. Budiantara and V. Ratnasari (2022). A Truncated Spline and Local Linear Mixed Estimator in Nonparametric Regression for Longitudinal Data and Its Application, Symmetry, 14 (12), 2687. DOI: 10.3390/sym14122687
[26] C. J. Stone (1980). Optimal rates of convergence for nonparametric estimators, Ann. Statist., 8, 1348–1360. DOI: 10.1214/aos/1176345206
[27] G. S. Watson (1964). Smooth regression analysis, Sankhyā, Ser. A, 26, 359–372. https://www.jstor.org/stable/25049340
[28] L. A. Zadeh (1965). Fuzzy sets, Inform. Control, 8, 338–353.
[29] S. Zhang and R. J. Karunamuni (1998). On kernel density estimation near endpoints, J. Stat. Plan. Infer., 70 (2), 301–316. DOI: 10.1016/S0378-3758(97)00187-0
[30] S. Zhang, R. J. Karunamuni and M. C. Jones (1999). An Improved Estimator of the Density Function at the Boundary, J. Am. Stat. Assoc. 94 (448), 1231–1240. DOI: 10.1080/01621459.1999.10473876
Published
2024-06-10
How to Cite
Fajardo, J. A. (2024). Boundary Estimation with the Fuzzy Set Regression Estimator. Divulgaciones Matemáticas, 82-106. Retrieved from https://produccioncientificaluz.org/index.php/divulgaciones/article/view/42240
Section
Research papers
Copyright (c) 2023 Jesús A. Fajardo
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