Conditionally Unbiased Bounded Influence Estimation in General Regression

Iu this paper we study robust estimation in general models for the dependence of a response y on an explanatory vector z. We extend previous work on bounded influence estimators in linear regression. Second we construct optimal bounded influence estimators for generalized linear models. We consider the class of estimators defined by an estimating equation with a conditionally unbiased score flwction given the desgin. The resulting estimators are said to be conditionally Fisherconsistent. Ordinary least squares in linear regression has this property as does the Mallows type bounde.d influence estimator. The Schweppe class does not have a conditionally unbiased score function if the errors are asymmetric. For generalized linear models, the optimal conditionally Fisher-consistent estimators are computationally simpler than the unconditional ones proposed by Stefanski, Carroll and Ruppert (1986) because the centering constant can be given in explicit form. The optimal score function contains an unknown auxiliary nuisance matrix B. In contrast to the estimator of Stefanski et al. (1986) estimation of B has in our case asymptotically no effect on the distribution of the estimator. Two examples using logistic regression are discussed in detail. It is shown that robust estimation can identify outliers even in situations where the model is close to being indeterminate.