Continuity and differentiability of regression M functionals

TL;DR

This paper investigates the mathematical properties of regression M functionals, including their continuity and differentiability, to establish conditions for their consistency and asymptotic normality, especially for high breakdown estimators like S and MM estimates.

Contribution

It introduces the concept of weak differentiability for regression M functionals and extends asymptotic normality results to more general dependent data conditions.

Findings

Regression MM-estimates are asymptotically normal under $\,\phi$-mixing conditions.

The paper establishes Fisher-consistency and robustness of the estimators.

It broadens the theoretical understanding of high breakdown regression estimators.

Abstract

This paper deals with the Fisher-consistency, weak continuity and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. A restricted type of differentiability, called weak differentiability, is defined, which suffices to prove the asymptotic normality of estimates based on the functionals. This approach allows to prove the consistency, asymptotic normality and qualitative robustness of M estimates under more general conditions than those required in standard approaches. In particular, we prove that regression MM-estimates are asymptotically normal when the observations are $\phi$-mixing.