The Influence Curve and Its Role in Robust Estimation

TLDR

Robust estimation near strict parametric models is briefly sketched and applied to classical situations. The paper investigates the first derivative of an estimator as a functional to study local robustness properties. It relates von Mises functionals, the jackknife, and U‑statistics, and discusses classical and new estimators such as trimmed and Winsorized means, Huber estimators, and general M‑estimators. A table of 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.