Functional stability of one-step GM-estimators in approximately linear regression

Abstract

This paper provides a comparative sensitivity analysis of one-step Newton–Raphson estimators for linear regression. Such estimators have been proposed as a way to combine the global stability of high breakdown estimators with the local stability of generalized maximum likelihood estimators. We analyze this strategy, obtaining upper bounds for the maximum bias induced by $\varepsilon$-contamination of the model. These bounds yield break-down points and local rates of convergence of the bias as $\varepsilon$decreases to zero. We treat a unified class of Newton–Raphson estimators, including one-step versions of the well-known Schweppe, Mallows and Hill–Ryan GM estimators. Of the three well-known types, the Hill–Ryan form emerges as the most stable in terms of one-step estimation. The Schweppe form is susceptible to a breakdown of the Hessian matrix. For this reason it fails to improve on the local stability of the initial estimator, and it may lead to falsely optimistic estimates of precision.