A definition of qualitative robustness for general point estimators, and examples

TL;DR

This paper introduces a new definition of qualitative robustness for general point estimators, providing criteria applicable across various statistical models and connecting to Hampel robustness in nonparametric cases.

Contribution

It proposes a unified robustness definition for point estimators and applies it to parametric, semiparametric, and nonparametric models, clarifying robustness classification.

Findings

The new robustness definition encompasses Hampel robustness in nonparametric models.

Criteria for robustness are established and demonstrated across different models.

Plug-in estimators are classified based on their degrees of robustness.

Abstract

A definition of qualitative robustness for point estimators in general statistical models is proposed. Some criteria for robustness are established and applied to estimators in parametric, semiparametric, and nonparametric models. In specific nonparametric models, the proposed definition boils down to Hampel robustness. It is also explained how plug-in estimators in certain nonparametric models can be reasonably classified w.r.t. their degrees of robustness.