Multivariate Dispersion, Central Regions, and Depth

de Karl Mosler

The lift zonoid approach is based on a new representation of probability measures: a d-variate probability measure is represented by a convex set, its lift zonoid. First, lift zonoids are useful in data analysis to describe an empiricaldistribution by central (so- called trimmed) regions. They give rise to a concept of data depth related to the mean which is also useful in nonparametric tests for location and scale. Second, for comparing random vectors, the set inclusion of lift zonoids defines a stochastic order that reflects the dispersion of random vectors. This has many applications to stochastic comparison problems in economics and other fields. This monograph ves the first account in book form of the theory of lift zonoids and demonstrates its usefulness in multivariate analysis. Chapter 1 offers the reader an informal introduction to basic ideas, Chapter 2 presents a comprehensive investigation into the theory. The remaining seven chapters treat various applications of the lift zonoid approach and may be separately studied. Readers are assumed to have a firm grounding in probability at the graduate level. Karl Mosler is Professor of Statistics and Econometrics at the University of Cologne. He is Editor of the Allgemeines Statistisches Archive, Journal of the German Statistical Society, and has authored numerous research articles and four books (all with Springer-Verlag) in statistics and operations research.