Abstract
Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the so-called optimal bandwidth h n = O(|n|−1/5), n ∈ Z d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O(|n|−4/5 log |n|). Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.
This is a preview of subscription content, access via your institution.
References
Fan, J., Gijbels, I.: Local polynomial modelling and its applications, Chapman and Hall, London, 1996
Fan, J., Jiang, J.: Variable bandwidth and one-step local M-estimator. Sci. China Ser. A, 43, 65–81 (2000)
Jiang, J., Mack, Y. P.: Robust local polynomial regression for dependent data. Statist. Sinica, 11, 705–722 (2001)
Cai, Z., Ould-Saïd, E.: LocalM-estimator for nonparametric time series. Statist. Probab. Lett., 65, 433–449(2003)
Lin, L.: Robust depth-weighted wavelet for nonparametric regression models. Acta Mathematica Sinica, English Series, 21(3), 585–592 (2005)
Tran, L.T.: Kernel density estimation on random fields. J. Multi. Analy., 34, 37–53 (1990)
Hallin, M., Lu, Z., Tran, L.T.: Local linear spatial regression. Ann. Statist., 32, 2469–2500 (2004)
Lu, Z., Chen, X.: Spatial kernel regression estimation: weak consistency. Statist. Probab. Lett., 68, 125–136 (2004)
Lin, Z. Y., Li, D. G., Gao, J. T.: Local linear M-estimators for spatial processes, Technical report available at www.maths.uwa.edu.au/:_jiti/llg.pdf, 2007
Hong, S. Y.: Bahadur representation and its applications for local polynomial estimates in nonparametric M-estimation. J. Nonparametric Statist., 52, 237–251 (2003)
Cheng, Y., Gooijer, J.: Bahadur representation for the nonparametric M-estimator under α-mixing dependence. Tinbergen Institute Discussion Paper, Department of Quantitative Economics, Faculty of Economics and Econometrics, University of Amsterdam, 2005
Pham, T. D., Tran, L. T.: Some strong mixing properties of time series models. Stoch. Proc. Their Appl., 19, 297–303 (1985)
Yao, Q. W.: Exponential inequalities for spatial processes and uniform convergence rates for density estimation, In Development of modern Statistics and Related Topics — In Celebration of Prof. Yaoting Zhang’s 70th Birthday, Zhang, H. and Huang, J. (edit.), World Scientific, Singapore, 118–128, 2003
Lee, Y. K., Choi, H., Park, B. U., Yu, K. S.: Local likelihood density estimation on random fields. Statist. Probab. Lett., 68, 347–357 (2004)
Niemiro, W.: Asymptotics for M-estimators defined by convex minimization. Ann. Statist., 20, 1514–1533 (1992)
Ortega, J. M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (No. 10771192)
Rights and permissions
About this article
Cite this article
Chen, J., Li, D.G. & Zhang, L.X. Bahadur representation of nonparametric M-estimators for spatial processes. Acta. Math. Sin.-English Ser. 24, 1871–1882 (2008). https://doi.org/10.1007/s10114-008-6589-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-008-6589-2