Asymptotic Statistical Properties of Redescending M-estimators in Linear Models with Increasing Dimension

TL;DR

This paper investigates the asymptotic properties of a broad class of redescending M-estimators in high-dimensional linear models, establishing conditions for consistency and asymptotic normality as the number of covariates grows.

Contribution

It extends existing results to include high breakdown point estimators like S- and MM-estimators in increasing dimension settings.

Findings

Proves consistency under p/n → 0

Establishes asymptotic normality if p^3/n → 0

Includes popular high breakdown point estimators

Abstract

This paper deals with the asymptotic statistical properties of a class of redescending M-estimators in linear models with increasing dimension. This class is wide enough to include popular high breakdown point estimators such as S-estimators and MM-estimators, which were not covered by existing results in the literature. We prove consistency assuming only that $p/n \rightarrow 0$ and asymptotic normality essentially if $p^{3}/n \rightarrow 0$, where $p$ is the number of covariates and $n$ is the sample size.