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.
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Supported by National Natural Science Foundation of China (No. 10771192)
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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