Estimating the model of the majority of the data

In the last years attention has been devoted to the construction of estimators that (optimally) bound the bias (and/or the variance) under the assumption that the data are in a neighborhood of the assumed model. These estimators are often referred to as able to recognize the model of the majority of the data under contamination. We formalize the concept of the “model of the majority of the data” and we study the conditions under which this model is unique. We provide simple examples where minimum max bias estimators fail to recognize the model of the majority of the data. A quantitative measure of robustness is introduced with the aim of evaluating the estimators regarding this aspect. We define estimators which are asymptotically optimal with respect to this criterion. For completeness of the presentation, we study their maxbias functions, breakdowns and asymptotic distributions. Simulations and examples are provided throughout to illustrate the theoretical results and the performance of the estimators.