ABSTRACT

As we have emphasized in the previous chapters, the primary motivation for developing the minimum distance estimators considered in this book is to generate a class of estimators which combine full asymptotic efficiency with strong robustness features. However, we have also pointed out that the estimators within our class of interest match the maximum likelihood estimator in terms of their influence functions at the model, and hence they are not first-order robust in that sense, although they are first-order efficient. This result notwithstanding, the robustness credentials of the minimum Hellinger distance estimator and many other minimum distance estimators within the class of disparities are undeniable. However, to the followers of the classical robustness theory some explanations about the sources of robustness of these minimum distance estimators are necessary. In this chapter we take on this robustness issue, and discuss in detail several robustness indicators including graphical interpretations, α-influence functions, second-order bias approximations, breakdown analysis and contamination envelopes. This comprehensive description establishes that the minimum distance estimators in this class are genuine competitors to the classical estimators present in the robustness literature. For instance, Tamura and Boos (1986) showed that the affine invariant minimum Hellinger distance estimator for multivariate location and scatter has a breakdown point of at least 14 , a quantity independent of the data dimension k; in contrast, the breakdown point of the affine invariant M -estimator is at most 1/(k + 1). Donoho and Liu (1988a) showed that the minimum Hellinger distance estimator has the best stability against Hellinger contamination among all Fisher consistent functionals. In real data situations, many of our minimum distance estimators behave like bounded influence estimators for all practical purposes. All in all, as our presentation in this chapter will show, the minimum distance estimators within the class of disparities are solid and useful tools for the applied statistician and researchers in other

Approach

We have already introduced the residual adjustment function A(δ) of the disparity in connection with the minimum disparity estimating equations in Chapter 2; see Equations (2.37), (2.38) and (2.39). In that chapter we used the RAF extensively to describe the estimation procedure based on the minimization of the corresponding measure. In order to emphasize the robustness concept, and to describe the tradeoff between robustness and efficiency, we summarize the following points about the residual adjustment function of a disparity.