Robust Analysis of M-Estimators of Nonlinear Models

Estimation of nonlinear models findsapplications in every field of engineering andthe sciences. Much work has been done tobuild solid statistical theories for its use andinterpretation. However, there has been littleanalysis of the tolerance of nonlinear modelestimators to deviations from assumptionsand normality.We focus on analyzing the robustnessproperties of M-estimators of nonlinearmodels by studying the effects of deviationsfrom assumptions and normality on theseestimators. We discuss St. Laurent andCook's Jacobian Leverage and identify therelationship of the technique to therobustness concept of influence. We deriveinfluence functions for M-estimators ofnonlinear models and show that influence ofposition becomes, more generally, influenceof model. The result shows that, forM-estimators, we must bound not onlyinfluence of residual but also influence ofmodel. Several examples highlight the uniqueproblems of nonlinear model estimation anddemonstrate the utility of the influencefunction.