The trimmed mean in non-parametric regression function estimation

Dhar, Subhra; Jha, Prashant; Rakshit, Prabrisha

doi:10.1090/tpms/1174

Authors: Subhra Sankar Dhar, Prashant Jha and Prabrisha Rakshit
Journal: Theor. Probability and Math. Statist. 107 (2022), 133-158
MSC (2020): Primary 62G08; Secondary 62G05
DOI: https://doi.org/10.1090/tpms/1174
Published electronically: November 8, 2022
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article studies a trimmed version of the Nadaraya–Watson estimator for the unknown non-parametric regression function. The characterization of the estimator through the minimization problem is established, and its pointwise asymptotic distribution is derived. The robustness property of the proposed estimator is also studied through the breakdown point. Moreover, similar to the trimmed mean in the location model, and for a wide range of trimming proportion, the proposed estimator possesses good efficiency and high breakdown point, which is out of the ordinary properties for any estimator. Furthermore, the usefulness of the proposed estimator is shown for two benchmark real data and various simulated data.

References

Ibrahim M. Almanjahie, Mohammed Kadi Attouch, Omar Fetitah, and Hayat Louhab, Robust kernel regression estimator of the scale parameter for functional ergodic data with applications, Chil. J. Stat. 11 (2020), no. 2, 73–93. MR 4206207
Mohammed Attouch, Ali Laksaci, and Elias Ould-Saïd, Asymptotic distribution of robust estimator for functional nonparametric models, Comm. Statist. Theory Methods 38 (2009), no. 8-10, 1317–1335. MR 2538143, DOI 10.1080/03610920802422597
Mohammed Attouch, Ali Laksaci, and Elias Ould Saïd, Asymptotic normality of a robust estimator of the regression function for functional time series data, J. Korean Statist. Soc. 39 (2010), no. 4, 489–500. MR 2780220, DOI 10.1016/j.jkss.2009.10.007
Nadjia Azzedine, Ali Laksaci, and Elias Ould-Saïd, On robust nonparametric regression estimation for a functional regressor, Statist. Probab. Lett. 78 (2008), no. 18, 3216–3221. MR 2479480, DOI 10.1016/j.spl.2008.06.018
Peter J. Bickel, On some robust estimates of location, Ann. Math. Statist. 36 (1965), 847–858. MR 177484, DOI 10.1214/aoms/1177700058
Patrick Billingsley, Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 534323
Graciela Boente and Ricardo Fraiman, Robust nonparametric regression estimation for dependent observations, Ann. Statist. 17 (1989), no. 3, 1242–1256. MR 1015148, DOI 10.1214/aos/1176347266
Graciela Boente and Ricardo Fraiman, Asymptotic distribution of robust estimators for nonparametric models from mixing processes, Ann. Statist. 18 (1990), no. 2, 891–906. MR 1056342, DOI 10.1214/aos/1176347631
Graciela Boente and Ricardo Fraiman, Local $L$-estimators for nonparametric regression under dependence, J. Nonparametr. Statist. 4 (1994), no. 1, 91–101. MR 1366366, DOI 10.1080/10485259408832603
Graciela Boente and Alejandra Vahnovan, Strong convergence of robust equivariant nonparametric functional regression estimators, Statist. Probab. Lett. 100 (2015), 1–11. MR 3324068, DOI 10.1016/j.spl.2015.01.028
Graciela Boente and Alejandra Vahnovan, Robust estimators in semi-functional partial linear regression models, J. Multivariate Anal. 154 (2017), 59–84. MR 3588557, DOI 10.1016/j.jmva.2016.10.005
Pavel Čížek, Generalized method of trimmed moments, J. Statist. Plann. Inference 171 (2016), 63–78. MR 3458068, DOI 10.1016/j.jspi.2015.11.004
P. Čížek, J. Tamine, and W. Härdle, Smoothed $L$-estimation of regression function, Comput. Statist. Data Anal. 52 (2008), no. 12, 5154–5162. MR 2526582, DOI 10.1016/j.csda.2008.05.024
R. M. Clark, Non-parametric estimation of a smooth regression function, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 1, 107–113. MR 494661, DOI 10.1111/j.2517-6161.1977.tb01611.x
Gérard Collomb and Wolfgang Härdle, Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations, Stochastic Process. Appl. 23 (1986), no. 1, 77–89. MR 866288, DOI 10.1016/0304-4149(86)90017-7
C. Crambes, L. Delsol, and A. Laksaci, Robust nonparametric estimation for functional data, J. Nonparametr. Stat. 20 (2008), no. 7, 573–598. MR 2454613, DOI 10.1080/10485250802331524
Luc Devroye, Laws of the iterated logarithm for order statistics of uniform spacings, Ann. Probab. 9 (1981), no. 5, 860–867. MR 628878
Subhra Sankar Dhar, Trimmed mean isotonic regression, Scand. J. Stat. 43 (2016), no. 1, 202–212. MR 3467002, DOI 10.1111/sjos.12173
Subhra Sankar Dhar and Probal Chaudhuri, A comparison of robust estimators based on two types of trimming, AStA Adv. Stat. Anal. 93 (2009), no. 2, 151–158. MR 2511592, DOI 10.1007/s10182-008-0099-5
Subhra Sankar Dhar and Probal Chaudhuri, On the derivatives of the trimmed mean, Statist. Sinica 22 (2012), no. 2, 655–679. MR 2954356, DOI 10.5705/ss.2010.155
J. Fan and I. Gijbels, Local polynomial modelling and its applications, Monographs on Statistics and Applied Probability, vol. 66, Chapman & Hall, London, 1996. MR 1383587
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
Luis Angel García-Escudero, Alfonso Gordaliza, and Carlos Matrán, Trimming tools in exploratory data analysis, J. Comput. Graph. Statist. 12 (2003), no. 2, 434–449. MR 1983163, DOI 10.1198/1061860031806
Luis A. García-Escudero, Alfonso Gordaliza, Carlos Matrán, and Agustin Mayo-Iscar, A general trimming approach to robust cluster analysis, Ann. Statist. 36 (2008), no. 3, 1324–1345. MR 2418659, DOI 10.1214/07-AOS515
Theo Gasser and Hans-Georg Müller, Kernel estimation of regression functions, Smoothing techniques for curve estimation (Proc. Workshop, Heidelberg, 1979) Lecture Notes in Math., vol. 757, Springer, Berlin, 1979, pp. 23–68. MR 564251
Robert V. Hogg, Some observations on robust estimation, J. Amer. Statist. Assoc. 62 (1967), 1179–1186. MR 221630, DOI 10.1080/01621459.1967.10500924
Peter J. Huber, Robust statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981. MR 606374, DOI 10.1002/0471725250
Louis A. Jaeckel, Some flexible estimates of location, Ann. Math. Statist. 42 (1971), 1540–1552. MR 350951, DOI 10.1214/aoms/1177693152
Jana Jurečková, Roger Koenker, and A. H. Welsh, Adaptive choice of trimming proportions, Ann. Inst. Statist. Math. 46 (1994), no. 4, 737–755. MR 1325993, DOI 10.1007/BF00773479
Jana Jurečková and Bohumír Procházka, Regression quantiles and trimmed least squares estimator in nonlinear regression model, J. Nonparametr. Statist. 3 (1994), no. 3-4, 201–222. MR 1291545, DOI 10.1080/10485259408832583
H. Kaya, Pm. Tüfekci, and F. S. Gürgen, Local and global learning methods for predicting power of a combined gas & steam turbine, Proceedings of the International Conference on Emerging Trends in Computer and Electronics Engineering ICETCEE, 2012, pp. 13–18.
È. A. Nadaraja, On non-parametric estimates of density functions and regression, Teor. Verojatnost. i Primenen. 10 (1965), 199–203 (Russian, with English summary). MR 0172400
Ch.-H. Park, S. Lee, and J.-H. Chang, Robust closed-form time-of-arrival source localization based on $\alpha$-trimmed mean and Hodges–Lehmann estimator under NLOS environments, Signal Processing 111 (2015), 113–123.
M. B. Priestley and M. T. Chao, Non-parametric function fitting, J. Roy. Statist. Soc. Ser. B 34 (1972), 385–392. MR 331616, DOI 10.1111/j.2517-6161.1972.tb00916.x
Peter J. Rousseeuw and Annick M. Leroy, Robust regression and outlier detection, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1987. MR 914792, DOI 10.1002/0471725382
Robert J. Serfling, Approximation theorems of mathematical statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595165, DOI 10.1002/9780470316481
B. W. Silverman, Density estimation for statistics and data analysis, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1986. MR 848134, DOI 10.1007/978-1-4899-3324-9
Stephen M. Stigler, The asymptotic distribution of the trimmed mean, Ann. Statist. 1 (1973), 472–477. MR 359134
A. Tsanas, M. A. Little, P. E. McSharry, and L. O. Ramig, Accurate telemonitoring of Parkinson’s disease progression by noninvasive speech tests, IEEE Transactions on Biomedical Engineering 57 (2009), no. 4, 884–893.
P. Tüfekci, Prediction of full load electrical power output of a base load operated combined cycle power plant using machine learning methods, International Journal of Electrical Power & Energy Systems 60 (2014), 126–140.
Wei Wang, Nan Lin, and Xiang Tang, Robust two-sample test of high-dimensional mean vectors under dependence, J. Multivariate Anal. 169 (2019), 312–329. MR 3875602, DOI 10.1016/j.jmva.2018.09.013
Geoffrey S. Watson, Smooth regression analysis, Sankhyā Ser. A 26 (1964), 359–372. MR 185765
A. H. Welsh, The trimmed mean in the linear model, Ann. Statist. 15 (1987), no. 1, 20–45. With discussion by P. J. de Jongh, T. de Wet and Roger Koenker and a reply by Welsh. MR 885722, DOI 10.1214/aos/1176350248