Almost sure uniform convergence rates for M-smoothers with non-monotone score functions

Almost sure uniform convergence rates for M-smoothers have been obtained by Härdle, Janssen and Serfling (1988) in the case of bounded, continuous and monotone score functions. In this paper, we show that if there is an initial estimator whose error is a.s. uniformly bounded, then the same rate result holds without the boundedness assumption on the score function [psi](y, t) and with the assumptions of continuity and monotonicity in t replaced by a.s. continuity at each t and a.s. piecewise monotonicity. Since such initial estimators are easy to construct in most situations, this considerably enlarges the scope of the rate result.