Rate of Convergence of One- and Two-Step $M$-Estimators with Applications to Maximum Likelihood and Pitman Estimators

Abstract

A one-step version $M^{(1)}_n$ and a two-step version $M^{(2)}_n$ of a general $M$-estimator $M_n$ are suggested such that $M_n - M^{(1)}_n = O_p(n^{-1})$ and $M_n - M^{(2)}_n = O_p(n^{-3/2})$ for every $n^{1/2}$-consistent initial estimator and under some regularity conditions. In the special case of maximum likelihood estimation, this among other yields that the second-order efficiency properties of $M^{(2)}_n$ coincide with those of $M_n$. An application to the Pitman estimator of location is considered.