Bahadur Representation of Nonparametric M-Estimators for Spatial Processes

Under some mild conditions,we establish a strong Bahadur representation of a generalclass of nonparametric local linear M-estimators for mixing processes on a random field.If the so-called optimal bandwidth h=O(|n|,n ∈ Z~d,is chosen,then the remainder rates in the Bahadurrepresentation for the local M-estimators of the regression function and its derivative are of orderO(|n|log|n|).Moreover,we derive some asymptotic properties for the nonparametric local linearM-estimators as applications of our result.