Domains of weak continuity of statistical functionals with a view toward robust statistics

TL;DR

This paper investigates the weak and local robustness properties of statistical estimators, including maximum likelihood estimators, by identifying relevant distribution sets and extending the theory of robust estimation with new continuity conditions.

Contribution

It introduces the concept of local robustness sets for estimators and extends the robust estimation theory with a Hampel-type theorem linking local robustness to continuity.

Findings

Maximum likelihood estimators are robust on their natural parametric domains.

Many estimators satisfy weak or local robustness properties under certain distribution sets.

The paper extends the theory of robust estimation with a new continuity-based Hampel-type theorem.

Abstract

Many standard estimators such as several maximum likelihood estimators or the empirical estimator for any law-invariant convex risk measure are not (qualitatively) robust in the classical sense. However, these estimators may nevertheless satisfy a weak (Kr\"atschmer/Schied/Z\"ahle 2012, 2014) or a local (Z\"ahle 2016) robustness property on relevant sets of distributions. One aim of our paper is to identify sets of local robustness, and to explain the benefit of the knowledge of such sets. For instance, we will be able to demonstrate that many maximum likelihood estimators are robust on their natural parametric domains. A second aim consists in extending the general theory of robust estimation to our local framework. In particular we provide a corresponding Hampel-type theorem linking local robustness of a plug-in estimator with a certain continuity condition.