Characterizing M-estimators

TL;DR

This paper provides a comprehensive characterization of M-estimators in semiparametric models, linking forecast evaluation loss functions with estimation theory to enable advanced research and application.

Contribution

It introduces a novel theoretical framework connecting loss functions from forecast evaluation with M-estimation in semiparametric models.

Findings

Unified characterization of M-estimators and loss functions

Enables leveraging forecast evaluation results in estimation theory

Facilitates research on efficient and equivariant M-estimation

Abstract

We characterize the full classes of M-estimators for semiparametric models of general functionals by formally connecting the theory of consistent loss functions from forecast evaluation with the theory of M-estimation. This novel characterization result opens up the possibility for theoretical research on efficient and equivariant M-estimation and, more generally, it allows to leverage existing results on loss functions known from the literature of forecast evaluation in estimation theory.