A robust combined M-estimator for the negative binomial model
Robust inference for overdispersed count data is crucial in applications where outliers may substantially distort classical likelihood-based estimation of both the mean and dispersion. We develop robust estimation procedures for independent and identically distributed negative binomial data and provide practical guidelines on the choice of the estimator. We propose a combined robust M-estimator that jointly estimates the mean and dispersion parameter through an alternating updating scheme based on bounded score functions. The mean update relies on a bias-corrected Tukey-type M-estimator, extending the Poisson framework of Elsaied and Fried (2016) to the negative binomial setting, while the dispersion update follows a robust modification of score-based estimation in the spirit of Aeberhard et al. (2014). Under standard regularity conditions, we establish key theoretical guarantees for the proposed estimator, including consistency, local convergence of the alternating algorithm, asymptotic normality, and robustness to outliers through bounded influence. Extensive simulation studies compare the proposed method with maximum likelihood estimation, minimum disparity estimators, and weighted maximum likelihood approaches under both clean and contaminated sampling. The results show that classical likelihood procedures can be highly sensitive to additive contamination, particularly in dispersion estimation, whereas the proposed estimator achieves a favorable efficiency–robustness trade-off across a broad range of sample sizes and contamination regimes, while retaining high efficiency under the nominal model. An analysis of epileptic seizure count data further illustrates the practical relevance of the approach and the stability of the resulting inference without relying on ad hoc data cleaning.