Asymptotic representation of m-estimators and rate of convergence of one-step m-estimators for parametric models with shrinking contamination

Abstract

Summary. Let Tnbe an M-estimator of a parameter [d], generated by a function [d]. It is shown under some conditions on [d] and the model {F[d]}[d]=[d] that the formula [d]n(Tn- [d]n+ (yn)-1[d]i=n i=n[d] (x, [d]n))= OG[d] n n(n-1/2holds for Gnand [d]nfrom an infinitesimal neighbourhood of F[d] and [d] respectively, for a constant y = y([d], F[d]). It is aiso considered a one-step version M(1) nof Tnsuch that, under the same conditions, Tn- M(1) n= OG[d] n n(n-1for every n1/2strong consistent initial estimator