Properties of robust M-estimators for Poisson and Negative Binomial data

We investigate robust M-estimators of location and over-dispersion for independent and identically distributed samples from Poisson and Negative Binomial (NB) distributions. We focus on asymptotic and small-sample efficiencies, outlier-induced biases, and biases caused by model mis-specification. This is important information for assessing the practical utility of the estimation method. Our results demonstrate that reasonably efficient estimation of location and over-dispersion parameters for count data is possible with samples sizes as small as n = 25. The sensitivity of these estimators to large outliers can be reduced compared to maximum likelihood estimators, especially when the amount of over-dispersion is small. We also conclude that serious biases result when using robust Poisson M-estimation with NB data, The biases are less serious when using robust NIB M-estimation with Poisson data.