Uniform bahadur representation for local polynomial estimates of M-regression and its application to the additive model

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Y-i, (X) under bar (i))}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.