The Influence Curve and its Role in Robust Estimation

TLDR

The paper examines the first derivative of an estimator as a functional to analyze local robustness properties. The authors sketch a theory of robust estimation near strict parametric models and apply it to classical situations. They relate von Mises functionals to jackknife and U‑statistics and discuss classical and novel estimators such as trimmed and Winsorized means, Huber, MLE, and M‑estimators. A table summarizing numerical robustness properties is presented.

Abstract

Abstract This paper treats essentially the first derivative of an estimator viewed as functional and the ways in which it can be used to study local robustness properties. A theory of robust estimation “near” strict parametric models is briefly sketched and applied to some classical situations. Relations between von Mises functionals, the jackknife and U-statistics are indicated. A number of classical and new estimators are discussed, including trimmed and Winsorized means, Huber-estimators, and more generally maximum likelihood and M-estimators. Finally, a table with some numerical robustness properties is given.