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Projection estimators for generalized linear models. (English) Zbl 1232.62085
Summary: We introduce a new class of robust estimators for generalized linear models which is an extension of the class of projection estimators for linear regression. These projection estimators are defined using an initial robust estimator for a generalized linear model with only one unknown parameter. We found a bound for the maximum asymptotic bias of the projection estimator caused by a fraction $$\varepsilon$$ of outlier contamination. For small $$\varepsilon$$, this bias is approximately twice the maximum bias of the initial estimator independently of the number of regressors. Since these projection estimators are not asymptotically normal, we define one-step weighted M-estimators starting at the projection estimators. These estimators have the same asymptotic normal distribution as the M-estimator and a degree of robustness close to the one of the projection estimator. We perform a Monte Carlo simulation for the case of binomial and Poisson regression with canonical links. This study shows that the proposed estimators compare favorably with respect to other robust estimators. Supplemental material containing the proofs and the numerical algorithm used to compute the P-estimator is available online.

##### MSC:
 62H12 Estimation in multivariate analysis 62J12 Generalized linear models (logistic models) 62F35 Robustness and adaptive procedures (parametric inference) 62F12 Asymptotic properties of parametric estimators 65C05 Monte Carlo methods 62E20 Asymptotic distribution theory in statistics
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