Defining extremes and trimming by minimum covering sets

Extremes in a sample of random vectors from d are defined as the points on the boundary of the smallest of a class of convex sets which contains the sample. A corresponding trimmed sum, the sum of the vectors omitting layers of the extremes, is proposed as a robust multivariate location estimator. Representations for the distributions of these quantities are derived and applied to give necessary and sufficient conditions for the consistency and asymptotic normality of the trimmed sum when the number of points removed is bounded in probability. Special cases of the methods are the minimum covering ellipse of a sample, and trimming by polyhedra.