Conditionally Unbiased Bounded-Influence Estimation in General Regression Models, with Applications to Generalized Linear Models

Abstract

Abstract In this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied. A subclass of the class of M estimators is defined by imposing the restriction that the score function must be conditionally unbiased, given x. Within this class of conditionally Fisher-consistent estimators, optimal bounded-influence estimators of regression parameters are identified, and their asymptotic properties are studied. The estimators studied in this article and the efficient bounded-influence estimators studied by Stefanski, Carroll, and Ruppert (1986) depend on an auxiliary centering constant and nuisance matrix. The centering constant can be given explicitly for the conditionally Fisher-consistent estimators, and thus they are easier to compute than the estimators studied by Stefanski et al. (1986). In addition, estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally Fisher-consistent estimators; the same is not true of the estimators studied by Stefanski et al. (1986). Logistic regression is studied in detail. The nature of influential observations in logistic regression is discussed, and two data sets are used to illustrate the methods proposed.