Asymptotic properties of one-step weighted $M$-estimators and applications to some regression problems

TL;DR

This paper investigates the asymptotic behavior of one-step weighted M-estimators for non-i.i.d. data, providing conditions for their normality and applications to nonlinear regression models for optimal estimation.

Contribution

It introduces explicit asymptotic analysis of one-step weighted M-estimators for diverse data distributions, extending their applicability to nonlinear regression models.

Findings

Established conditions for asymptotic normality.

Derived explicit asymptotically optimal estimators.

Applied results to nonlinear regression models.

Abstract

We study asymptotic behavior of one-step weighted $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent weighted $M$-estimators. Sufficient conditions are presented for asymptotic normality of the one-step weighted $M$-estimators under consideration. As a consequence, we consider some well-known nonlinear regression models where the procedure mentioned allow us to construct explicit asymptotically optimal estimators.