Minimum Hellinger Distance Estimation for the Analysis of Count Data

Abstract

Abstract Minimum Hellinger distance (MHD) estimation is studied in the context of discrete data. The MHD estimator is shown to provide an effective treatment of anomalous data points, and its properties are illustrated using short-term mutagenicity test data. Asymptotic normality for a discrete distribution with countable support is derived under a readily verified condition on the model. Breakdown properties of the MHD estimator and an outlier screen are compared. Count data occur frequently in statistical applications. For instance, in chemical mutagenicity studies, which comprise an important step in the identification of environmental carcinogens, much of the resultant data are counts. Woodruff, Mason, Valencia, and Zimmering (1984) reported anomalous counts in the sex-linked recessive lethal test in drosophila. These outliers can have a substantial impact on the experimental conclusions. MHD estimation provides a means for reliable inference when modeling count data that are prone to outliers. The MHD fit gives little weight to counts that are improbable relative to the model. On the other hand, the MHD estimator is asymptotically equivalent to the maximum likelihood estimator when the model is correct. This latter result, long known for a parametric multinomial model with finite support, is extended here to models with countable support. The breakdown point provides a quantification of outlier resistence. Roughly, it is the smallest proportion of outliers in the data that can cause an arbitrarily large shift in the estimate (Donoho and Huber 1983). Here the MHD estimator is shown to have an asymptotic breakdown point of ½ at the model.