We study the problem of nding an outlier-free subset of a set of points (or a probability distribution) in n-dimensional Euclidean space. A point x is de ned to be a -outlier if there exists some direction w in which its squared distance from the mean along w is greater than times the average squared distance from the mean along w[1]. Our main theorem is that for any > 0, there exists a (1 ) fraction of the original distribution that has no O( (b + log ))- outliers, improving on the previous bound of O(n b=). This bound is shown to be nearly the best possible. The theorem is constructive, and results in a 1 approximation to the following optimization problem: given a distribution (i.e. the ability to sample from it), and a parameter > 0, nd the minimum for which there exists a subset of probability at least (1 ) with no -outliers.

In investigations on the behaviour of robust estimators, typically their consistency and their asymptotic normality are studied as a necessity. Their rates of convergence, however, are often given less weight. We show here that the rate of convergence of a multivariate robust estimator to its true value plays an important role when using the estimator in procedures for identifying outliers in multivariate data. AMS 1991 Subject Classifications: Primary 62F35