Characterizing M-estimators.

This article focuses on characterizing the full classes of M-estimators for semiparametric models of general functionals by establishing a formal connection between consistent loss functions from forecast evaluation and M-estimation theory. The main result, Theorem 1, shows that a loss function is (strictly) model consistent for a parametric model if and only if it is (strictly) consistent for the target functional of the conditional distribution, under certain regularity and richness assumptions. This characterization enables leveraging known results on strictly consistent loss functions to identify all consistent M-estimators for various functionals such as means, quantiles, and expected shortfall. The article discusses implications for robust, efficient, equivariant, and Pareto-optimal M-estimation, highlighting limitations on robustness and equivariance properties of M-estimators and providing a theoretical foundation for efficiency results in semiparametric regression models.