kNN robustification equivariant nonparametric regression estimators for functional ergodic data

Year 2023, Volume: 52 Issue: 2, 512 - 528, 31.03.2023

Guenani Somia Bouabsa Wahiba Attouch Mohammed Kadi Fetitah Omar

https://doi.org/10.15672/hujms.1100871

Abstract

We discuss in this paper the robust equivariant nonparametric regression estimators for ergodic data with the k Nearst Neighbour (kNN) method. We consider a new robust regression estimator when the scale parameter is unknown. The principal aim is to prove the almost complete convergence (with rate) for the proposed estimator. Furthermore, a comparison study based on simulated data is also provided to illustrate the finite sample performances and the usefulness of the kNN approach and to prove the highly sensitive of the kNN approach to the presence of even a small proportion of outliers in the data.

Keywords

Functional data, ergodic data, kNN estimation, kernel estimate, uniform almost complete convergence rate, entropy

References

• [1] I. Almanjahie, K. Aissiri, A. Laksaci and Z. Chiker el Mezouar, The k nearest neighbors smoothing of the relative-error regression with functional regressor, Comm. Statist. Theory Methods 51 (12), 4196-4209, 2022.
• [2] I. Almanjahie, M. Attouch, Z. Kaid and H. Louab, Robust equivariant nonparametric regression estimators for functional ergodic data, Comm. Statist. Theory and Methods 50 (20), 3505-3521, 2020.
• [3] M. Attouch and W. Bouabsa, The k-nearest neighbors estimation of the conditional mode for functional data, Rev. Roumaine Math. Pures Appl. 58 (4), 393-415, 2013.
• [4] M. Attouch, W. Bouabsa and Z. Chiker el Mozouar, The k-nearest neighbors estimation of the conditional mode for functional data under dependency, International Journal of Statistics & Economics 19 (1), 48-60, 2018.
• [5] M. Attouch, A. Laksaci and E. Ould Saïd, Asymptotic distribution of robust estimator for functional non parametric models, Comm. Statist. Theory and Methods 45 (15), 230-287, 2009.
• [6] M. Attouch, A. Laksaci and E. Ould Saïd, Asymptotic normality of a robust estimator of the regression function for functional time series data, J. Korean Statist. Soc. 39 (22), 489-500, 2010.
• [7] M. Attouch, A. Laksaci and E. Ould Saïd, Robust regression for functional time series data, J. Jpn. Stat. Soc. 42 (2), 125-143, 2012.
• [8] N. Azzedine, A. Laksaci and E. Ould Saïd, On the robust nonparametric regression estimation for functional regressor, Statist. Probab. Lett. 78 (2), 3216-3221, 2008.
• [9] J. Beirlant, A. Berlinet and G. Biau, Higher order estimation at Lebesgue points, Ann. Inst. Statist. Math. 90 (60), 651-677, 2008.
• [10] F. Benziadi and A. Laksaci, Recursive kernel estimate of the conditional quantile for functional ergodic data, Comm. Statist. Theory and Methods 45 (12), 3097-3113, 2015.
• [11] G. Boente and R. Fraiman, Robust nonparametric regression estimation for dependent observations, Ann. Statist. 17 (5), 1242-1256, 1989.
• [12] G. Boente and A. Vahnovan, Strong convergence of robust equivariant nonparametric functional regression estimators, Statist. Probab. Lett. 100 (55), 1-11, 2015.
• [13] D. Bosq, Linear process in function spaces: theory and applications, Lecture Notes in Statistics 149, Springer-Verlag, 2000.
• [14] W. Bouabsa, Nonparametric relative error estimation via functional regressor by the k nearest neighbors smoothing under truncation random, Appl. Appl. Math. 16 (1), 97-116, 2021.
• [15] F. Burba, F. Ferraty and P. Vieu, k-nearest neighbour method in functional nonparametric regression, J. Nonparametr. Stat. 21 (4), 453-469, 2009.
• [16] J. Cheng and L. Zhang, Asymptotic properties of nonparametric M-estimation for mixing functional data, J. Statist. Plann. Inference 139 (70), 533-546, 2009.
• [17] G. Collomb, Estimation non paramétrique de la régression: revue bibliographique, Int. Stat. Rev. 49 (1), 75-93, 1981.
• [18] G. Collomb and H. Härdale, Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations, Stochastic Process. Appl. 23 (7), 77-89, 1979.
• [19] T.M. Cover, Estimation by the nearest neighbor rule, IEEE Trans. Inform. Theory 7 (14), 50-55, 1968.
• [20] C. Crambes and A. Laksaci, Robust nonparametric estimation for functional data, J. Nonparametr. Stat. 20 (5), 573-598, 2008.
• [21] L. Devroye, L. Györfi, A. Krzyzak and G. Lugosi, On the strong universal consistency of nearest neighbor regression function estimates, Ann. Statist. 20 (22), 1371-1385, 1994.
• [22] L. Devroye and T. Wagner, The strong uniform consistency of nearest neighbor density, Ann. Statist. 10 (5), 536-540, 1977.
• [23] L. Devroye and T. Wagner, Nearest neighbor methods in discrimination, in: Handbook of Statistics 2: Classification, Pattern Recognition and Reduction of Dimensionality, North-Holland, Amsterdam, 1982.
• [24] M. Ezzahrioui and E. Ould Saïd, Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data, J. Nonparametr. Stat. 20 (1), 3-18, 2008.
• [25] F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice, Springer-Verlag, New York, 2006.
• [26] G. Geenens, Curse of dimensionality and related issues in nonparametric functional regression, Stat. Surv. 5 (1), 30-43, 2011.
• [27] A. Gheriballah, A. Laksaci and S. Sekkal, Nonparametric M-regression for functional ergodic data, Statist. Probab. Lett. 83 (20), 902-908, 2013.
• [28] L. Györfi, Kohler, A. Krzyzak and H. Walk, A Distribution-Free Theory of Nonparametric Regression, Springer, New York, 2002.
• [29] P.J. Hurber, Robust estimation of a location parameter, in: S. Kotz and N.L. Johnson (ed.) Breakthroughs in Statistics, New York, Springer Science & Business Media, 1992.
• [30] Z. Kara, A. Laksaci and P. Vieu, Data-driven kNN estimation in nonparametric functional data analysis, J. Multivariate Anal. 153 (85), 176-188, 2017.
• [31] A.N. Kolmogorov and V. M. Tikhomirov, ε-entropy and ε-capacity of sets in function spaces, Uspekhi Mat. Nauk. 14 (2), 3-86, 1959.
• [32] N.L. Kudraszow and P. Vieu, Uniform consistency of kNN regressors for functional variables, Statist. Probab. Lett. 83 (26), 1863-1870, 2013.
• [33] N. Laïb and E. Ouled-Saïd, A robust nonparametric estimation of the autoregression function under an ergodic hypothesis, Canad. J. Statist. 28 (9), 817-828, 2000.
• [34] N. Laïb and D. Louani, Strong consistency of the regression function estimator for functional stationary ergodic data, J. Statist. Plann. Inference 141 (59), 359-372, 2011.
• [35] T. Laloë, A k-nearest neighbor approach for functional regression, Statist. Probab. Lett. 78 (10), 1189-1193, 2008.
• [36] J.P. Li, Strong convergence rates of error probability estimation in the nearest neighbor discrimination rule, J. Math. 15 (5), 113-118, 1985.
• [37] H. Lian, Convergence of functional k-nearest neighbor regression estimate with functional responses, Electron. J. Stat. 5 (2), 31-40, 2011.
• [38] N. Ling, S. Meng and P. Vieu, Uniform consistency rate of kNN regression estimation for functional time series data, J. Nonparametr. Stat. 31 (5), 451-468, 2019.
• [39] N. Ling and P. Vieu, Nonparametric modelling for functional data : selected survey and tracks for future, statistics, J. Nonparametr. Stat. 52 (4), 20-30, 2018.
• [40] D. Moore and J. Yackel, Consistency properties of nearest neighbor density function estimators, Ann. Statist. 20 (5), 143154, 1977.
• [41] S. Müller and J. Dippon, kNN kernel estimate for nonparametric functional regression in time series analysis, Technical Report, University of Stuttgart, Fachbereich Mathematik, 2011.
• [42] G.G. Roussas, Kernel estimates under association: strong uniform consistency, Statist. Probab. Lett. 12 (9), 215-224, 1991.
• [43] M. Samanta, Non-parametric estimation of conditional quantiles, Statist. Probab. Lett. 7 (5), 407-412, 1989
• [44] C.J. Stone, Optimal global rates of convergence for nonparametric regression, Ann. Statist. 10 (4), 1040-1053, 1982.
• [45] T. Tran, R. Wehrens and L. Buydens, kNN-kernel density-based clustering for highdimensional multivariate data, Comput. Statist. Data Anal. 51 (50), 513-525, 2006.