Bias-robust estimation of location and scale parameters

Abstract:

We begin by outlining the standard ideas of robust estimation of an unknown parameter (e.g., the mean of a normal distribution) under contamination by an unknown distribution. The basic concepts of ``contamination models'', ``asymptotic bias'' and ``bias robustness'' will be defined. Then a careful explanation will be given of Huber's 1964 classical result that the most bias-robust estimator of the centre of symmetry of a symmetric unimodal distribution is the median. This first part of the talk should be accessible to anyone who has had a first course in the basic concepts of statistics. The second part of the talk will be a bit more technical review of my recent extensions of Huber's bias-robustness results to problems of (i) scale parameter estimation and (ii) estimation of a location parameter in a model with unknown nuisance scale parameter. The key to these extensions is a close connection between the natural ``invariance'' required of the estimators and the ``unidentifiability'' of the parameters induced by the contamination of the models. This connection reduces the bias-robustness problem to a study of the geometry of a least favorable contaminating density. One new result is that the most bias-robust estimator (suitably defined in situation (ii)) of the centre of symmetry of a symmetric strongly unimodal distribution is not the median, but is the midpoint of the shortest $100gamma%$ of the distribution of the observations (where $gamma$ depends on the proportion of contamination).The results will be explained intuitively, with no proofs given during the talk.