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Conditionally unbiased bounded-influence estimation in general regression models, with applications to generalized linear models. (English) Zbl 0679.62024
Summary: 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 L. A. Stefanski, R. J. Carroll, and D. Ruppert [Biometrika 73, 413-424 (1986; Zbl 0616.62043)] 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. 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. 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.

MSC:
62F35 Robustness and adaptive procedures (parametric inference)
62J99 Linear inference, regression
62J05 Linear regression; mixed models
62F10 Point estimation
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