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.