The excess-mass ellipsoid

Abstract

The excess-mass ellipsoid is the ellipsoid that maximizes the difference between its probability content and a constant multiple of its volume, over all ellipsoids. When an empirical distribution determines the probability content, the sample excess-mass ellipsoid is a random set that can be used in contour estimation and tests for multimodality. Algorithms for computing the ellipsoid are provided, as well as comparative simulations. The asymptotic distribution of the parameters for the sample excess-mass ellipsoid are derived. It is found that a n13 normalization of the center of the ellipsoid and lengths of its axes converge in distribution to the maximizer of a Gaussian process with quadratic drift. The generalization of ellipsoids to convex sets is discussed.