A robust conditional maximum likelihood estimator for generalized linear models with a dispersion parameter

Highly robust and efficient estimators for generalized linear models with a dispersion parameter are proposed. The estimators are based on three steps. In the first step, the maximum rank correlation estimator is used to consistently estimate the slopes up to a scale factor. The scale factor, the intercept, and the dispersion parameter are robustly estimated using a simple regression model. Then, randomized quantile residuals based on the initial estimators are used to define a region S such that observations out of S are considered as outliers. Finally, a conditional maximum likelihood (CML) estimator given the observations in S is computed. We show that, under the model, S tends to the whole space for increasing sample size. Therefore, the CML estimator tends to the unconditional maximum likelihood estimator and this implies that this estimator is asymptotically fully efficient. Moreover, the CML estimator maintains the high degree of robustness of the initial one. The negative binomial regression case is studied in detail.