Robust Estimates of Location: Symmetry and Asymmetric Contamination

Abstract

The problem of finding location estimators which are "robust" against deviations from normality has received increasing attention in the last several years. See, for example, Tukey (1960), Huber (1968), and papers cited therein. In the theoretical work done on the estimation of a location parameter, the underlying distribution is usually assumed to be symmetric, and the estimand is taken to be the center of symmetry, a natural quantity to estimate in this situation. Since the finite sample size properties of many proposed estimators are difficult to study analytically, most research has focussed on their more easily ascertainable asymptotic properties, which, it is hoped, will provide useful approximations to the finite sample size case. Most of the estimators commonly studied are, under suitable regularity conditions, asymptotically normal about the center of symmetry, with asymptotic variance depending on the underlying distribution. We thus have a simple criterion, the asymptotic variance, for comparing the performance of different estimators for a given underlying distribution, and of a given estimator for different underlying distributions. Huber (1964) has formulated and solved some minimax problems, in which the estimators are judged by their asymptotic variance. In Section 2 we define and state the asymptotic variances which have been found for the three most commonly studied types of location estimators. In Section 3 we demonstrate some relationships among the three types of estimators, and in Section 4 we show that Huber's minimax result applies to all three types. Then, in Section 5 we consider an aspect of the more general estimation problem in which the distributions are not assumed symmetric. A model of asymmetric contamination of a symmetric distribution is formulated, in which the amount of asymmetry tends to zero as the sample size increases. The estimators here are thought of as estimating the center of the symmetric component of the distribution. The maximum likelihood type estimators are shown to be asymptotically normal under this model, but with a bias that tends to zero as the sample size increases. The estimators may be judged by their asymptotic mean squared error, a concept which is made meaningful by the model. We conclude in Section 6 with a minimax result analogous to Huber's, for which we allow both symmetric and asymmetric contamination of a given distribution and judge the estimators by their asymptotic mean squared error.