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This book introduces a new representation of probability measures, the lift zonoid representation, and demonstrates its usefulness in statistical applications.

The material divides into nine chapters.
Chapter 1 exhibits the main idea of the lift zonoid representation and surveys the principal results of later chapters without proofs.
Chapter 2 provides a thorough investigation into the theory of the lift zonoid. All principal properties of the lift zonoid are collected here for later reference. The remaining chapters present applications of the lift zonoid approach to various fields of multivariate analysis.
Chapter 3 introduces a family of central regions, the zonoid trimmed regions, by which a distribution is characterized. Its sample version proves to be useful in describing data.
Chapter 4 is devoted to a new notion of data depth, zonoid depth, which has applications in data analysis as well as in inference.
In Chapter 5 nonparametric multivariate tests for location and scale are investigated; their test statistics are based on notions of data depth, including the zonoid depth.
Chapter 6 introduces the depth of a hyperplane and tests which are built on it.
Chapter 7 is about volume statistics, the volume of the lift zonoid and the volumes of zonoid trimmed regions; they serve as multivariate measures of dispersion and dependency.
Chapter 8 treats the lift zonoid order, which is a stochastic order to compare distributions for their dispersion, and also indices and related orderings.
The final Chapter 9 presents further orderings of dispersion, which are particularly suited for the analysis of economic disparity and concentration.

The chapters are, to a large extent, self-contained. Cross-references between Chapters 3 to 9 have been kept to a minimum. A reader who wants to learn the theory of the lift zonoid approach may browse through the introductory survey in Chapter 1 and then study Chapter 2 carefully. A reader who is primarily interested in applications should read Chapter 1, proceed to any of the later chapters and go back to relevant parts of Chapter 1 for basic ideas and Chapter 2 for proofs and theoretical details when needed. Some standard notions from probability and convex analysis are found in Appendix A.

The research which is reported in this book started as a joint work with Gleb Koshevoy, Russian Academy of Sciences, in the mid-nineties, when he stayed with me in Hamburg and later in Cologne. Large parts of the manuscript are based on papers I have coauthored with him. So, I am heavily indebted to his ideas and scholarship.

I also thank Rainer Dyckerhoff for contributing the Chapter 5 on nonparametric statistical inference with data depths.

Several people have read parts of the manuscript.
I am very grateful to Jean Averous, Alfred Müller, and Wolfgang Weil, as well as to Katharina Cramer, Rainer Dyckerhoff, Richard Hoberg, and Thomas Möller for many helpful comments and hints to the literature.

Thanks are also to the Deutsche Forschungsgemeinschaft for funding Gleb Koshevoy's stays in Hamburg and Cologne.
Last not least I thank the editors of this series and John Kimmel of Springer Verlag for his continual encouragement and patience.

Cologne, Germany
Karl Mosler
February 2002