A data-based method for selecting tuning parameters in minimum distance estimators

A general method for selecting the tuning parameter in minimum distance estimators is proposed. The performance of this method, which involves minimising a data-based estimate of the asymptotic mean squared error function, is illustrated by its application to three different minimum distance estimators using simulated data. The choice of model family is subjective but information about the number and magnitude of outliers in the data is not needed because thereafter the method is entirely data-based. This approach is shown to optimise the performance of minimum distance methods by delivering estimators which are appropriate for the data. That is to say that, providing the correct family of models is chosen, the resulting estimators will be highly efficient when the method is applied to clean data and robust under contamination. Furthermore, utilising the asymptotic mean squared error function as a joint measure of robustness and efficiency in this way leads to a common framework for assessing the performance of many different minimum distance estimators. The relative merits of the three minimum distance estimators considered here are compared in detail and their performance, when the model family is normal, shown to rival that of the Huber estimator.