Weighted-mean regions: Theory and estimation Karl Mosler, Köln Université Libre Bruxelles, March 18, 2011 Abstract Weighted-mean regions are the level sets of a new class of depth functions, the weighted mean (WM) depth functions. They describe a probability distribution in Euclidean d-space regarding location, dispersion and shape, and they order given multivariate data with respect to their centrality. Also, they have a substantial interpretation in terms of multivariate set-valued risk measures that are coherent. The talk introduces the class of weighted-mean regions and their principal properties: affine equivariance, nestedness, continuity in the parameter as well as in the data, subadditivity and monotonicity. The notion is illustrated with several special cases, among them the zonoid regions and the ECH (expected convex hull) trimming. The weighted-mean regions of an empirical distribution are convex polytopes in Rd . A law of large numbers applies. Thus, given a sample, the empirical regions serve as natural estimates for the regions of the underlying probability distribution. In fact, the estimates can be computed for any dimension d by exact and approximate algorithms. They build on methods from computational geometry, by which the facets are characterized and their adjacency relations are found. If time allows, applications to multivariate risk measurement and stochastic linear programming may be discussed. The talk is based on joint work with Rainer Dyckerhoff and Pavel Bazovkin.