On the 'optimal' density power divergence tuning parameter

The density power divergence, indexed by a single tuning parameter alpha, has proved to be a very useful tool in minimum distance inference. The family of density power divergences provides a generalized estimation scheme which includes likelihood-based procedures (represented by choice for the tuning parameter) as a special case. However, under data contamination, this scheme provides several more stable choices for model fitting and analysis (provided by positive values for the tuning parameter alpha). As larger values of alpha necessarily lead to a drop in model efficiency, determining the optimal value of alpha to provide the best compromise between model-efficiency and stability against data contamination in any real situation is a major challenge. In this paper, we provide a refinement of an existing technique with the aim of eliminating the dependence of the procedure on an initial pilot estimator. Numerical evidence is provided to demonstrate the very good performance of the method. Our technique has a general flavour, and we expect that similar tuning parameter selection algorithms will work well for other M-estimators, or any robust procedure that depends on the choice of a tuning parameter.