The Threshold Breakdown Point

TL;DR

This paper introduces the threshold breakdown point and finite sample m-sensitivity to assess robustness of estimators, extending traditional breakdown concepts with new measures, theoretical analysis, and practical applications.

Contribution

It proposes novel robustness measures for finite samples, derives their properties for common estimators, and develops an inferential framework with bootstrap methods.

Findings

Derived threshold breakdown point and m-sensitivity for M-estimators

Extended breakdown analysis to hypothesis testing and power functions

Validated methods through numerical examples and blood pressure data application

Abstract

We introduce a novel approach to finite sample robustness that avoids the pessimism of traditional breakdown analyses. We define the threshold breakdown point, the smallest contamination fraction needed to induce a prescribed deviation, and the finite sample m-sensitivity, the worst-case deviation that an estimator can incur after m observations are contaminated. We derive these measures for commonly used M-estimators, their standard errors and related test statistics. This allows us to extend the decision breakdown point of Zhang (1996) to obtain general breakdown characterizations for hypothesis testing, and show how these notions correspond to finite sample counterparts of the power and level breakdown functions of He, Simpson and Portnoy (1990). We complement our work with an inferential framework for the threshold breakdown and m-sensitivity that yields consistency and asymptotic normality results, as well as a valid multiplier bootstrap for uncertainty quantification. We illustrate the practical utility of our methods in various numerical examples and an application to a two sample testing problem for a blood pressure dataset.