Adaptive Nadaraya-Watson kernel regression estimators utilizing some non-traditional and robust measures: a numerical application of British food data

Abstract

In nonparametric regression research, estimation of regression function is a prime concern. Recently, researchers developed some modified Nadaraya-Watson (N-W) regression estimators utilizing robust mean, median and harmonic mean. In this paper, we propose to utilize the additive combination of non-traditional measures i.e. (Hodges-Lehmann, Mid-Range, Tri-Mean, Quartile-Deviation) with the robust minimum covariance determinant (MCD) scale estimator in (N-W) regression estimator. Utilizing these measures, we get some new versions of (N-W) regression estimator. We also attempted to derive the properties of the proposed versions, such as bias, variance, mean square error (MSE), and mean integrated square error (MISE). The proposed estimators are compared with some of the existing estimators available in literature through a simulation study, utilizing two artificial populations. We also incorporated real-life application by taking British food data set denoted as Engel95, and assess the predictive ability of nonparametric regression, based on proposed and existing N-W estimators.

Keywords

• Nonparametric regression
• N-W kernel estimator
• Robust measures

References

1. [1] I.S. Abramson, On bandwidth variation in kernel estimates-a square root law, Ann. Statist. 10 (4), 1217-1223, 1982.
2. [2] T.H. Ali, Modification of the adaptive Nadaraya-Watson kernel method for nonparametric regression (simulation study), Comm. Statist. Simulation Comput. 51 (2), 391-403, 2022.
3. [3] D.F Andrews, P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers and J.W. Tukey, Robust Estimates of Location: Survey and Advances, Princeton University Press, Princeton, New Jersey, 1972.
4. [4] G.R. Arce and S.A. Fontama, On the midrange estimator, IEEE Trans. Acoust., Speech, Signal Process. 36 (6), 920-922, 1988.
5. [5] S. Demir, Adaptive kernel density estimation with generalized least square crossvalidation, Hacet. J. Math. Stat. 48 (2), 616-625, 2019.
6. [6] S. Demir and O. Toktamis, On the adaptive nadaraya-watson kernel regression estimators, Hacet. J. Math. Stat. 39 (3), 429-437, 2010.
7. [7] A. Eftekharian and M. Razmkhah, On estimating the distribution function and odds using ranked set sampling, Statist. Probab. Lett. 122, 1-10, 2017.
8. [8] A. Eftekharian and H. Samawi, On kernel-based quantile estimation using different stratified sampling schemes with optimal allocation, J. Stat. Comput. Simul. 91 (5), 1040-1056, 2021.

9. [9] E.B. Ferrell, Control charts using midranges and medians, Industrial Quality Control 9 (5), 30-34, 1953.
10. [10] M. Hanif, S. Shahzadi, U. Shahzad and N. Koyuncu, On the adaptive Nadaraya- Watson kernel estimator for the discontinuity in the presence of jump size, Suleyman Demirel Univ. Fen Bilim. Enst. Derg. 22 (2), 511-520, 2018.
11. [11] T.P. Hettmansperger and J.W. McKean, Robust Nonparametric Statistical Methods, Arnold, London, 1998.
12. [12] J.L. Hodges and E.L. Lehmann, Estimates of location based on rank tests, Ann. Math. Statist. 34 (2), 598-611, 1963.
13. [13] H.A. Khulood and I.A. Lutfiah, Modification of the adaptive Nadaraya-Watson kernel regression estimator, Sci. Res. Essays 9 (22), 966-971, 2014.
14. [14] Q. Li and J.S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton University Press, 2007.
15. [15] P.R. Rider, The midrange of a sample as an estimator of the population midrange, J. Amer. Statist. Assoc. 52 (280), 537-542, 1957.
16. [16] P.J. Rousseeuw, Multivariate estimation with high breakdown point, in: W. Grossmann, G. Pflug, I. Vincze and W. Wertz (ed.) Mathematical Statistics and Applications, Reidel Publishing Company, Dordrecht, 283297, 1985.
17. [17] P.J. Rousseeuw and K. van Driessen, A fast algorithm for the minimum covariance determinant estimator, Technometrics 41 (3), 212-223, 1999.
18. [18] R.S. Saksena, A Hand Book of Statistics, Motilal Banarsidass, Delhi, 1981.
19. [19] B.W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986.
20. [20] M.P. Wand and M.C. Jones, Kernel Smoothing, CRC Press, 1994.
21. [21] T. Wang, Y. Li and H. Cui, On weighted randomly trimmed means, J. Syst. Sci. Complex. 20 (1), 47-65, 2007.
22. [22] E. Zamanzade and M. Vock, Variance estimation in ranked set sampling using a concomitant variable, Statist. Probab. Lett. 105, 1-5, 2015.
23. [23] E. Zamanzade and X. Wang, Improved nonparametric estimation using partially ordered sets, in: Statistical Methods and Applications in Forestry and Environmental Sciences, Springer, Singapore, 57-77, 2020.