Asymptotic properties of one-step $M$-estimators based on nonidentically distributed observations with applications to nonlinear regression problems

TL;DR

This paper investigates the asymptotic properties of one-step M-estimators derived from non-identically distributed data, extending Fisher's approach, and applies these results to nonlinear regression for explicit, optimal estimators.

Contribution

It introduces a generalized framework for one-step M-estimators with non-i.i.d. data and demonstrates their application to nonlinear regression problems.

Findings

Derived explicit asymptotic properties of one-step M-estimators.

Applied the methodology to construct asymptotically optimal estimators in nonlinear regression.

Provided procedures for initial estimator construction for one-step methods.

Abstract

We study asymptotic behavior of one-step $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent $M$-estimators. These estimators generalize Fisher's one-step approximations to consistent maximum likelihood estimators. As a consequence, we consider some nonlinear regression problems where the procedure mentioned allow us to construct explicit asymptotically optimal estimators. We also consider the problem of constructing initial estimators which are needed for one-step estimation procedures.