Multivariate Dispersion, Central Regions, and Depth

The Lift Zonoid Approach

  • Karl Mosler

Part of the Lecture Notes in Statistics book series (LNS, volume 165)

Table of contents

  1. Front Matter
    Pages i-x
  2. Karl Mosler
    Pages 1-24
  3. Karl Mosler
    Pages 25-78
  4. Karl Mosler
    Pages 79-104
  5. Karl Mosler
    Pages 105-131
  6. Karl Mosler
    Pages 165-179
  7. Karl Mosler
    Pages 181-206
  8. Karl Mosler
    Pages 207-228
  9. Back Matter
    Pages 259-295

About this book


This book introduces a new representation of probability measures, the lift zonoid representation, and demonstrates its usefulness in statistical applica­ tions. The material divides into nine chapters. Chapter 1 exhibits the main idea of the lift zonoid representation and surveys the principal results of later chap­ ters without proofs. Chapter 2 provides a thorough investigation into the theory of the lift zonoid. All principal properties of the lift zonoid are col­ lected here for later reference. The remaining chapters present applications of the lift zonoid approach to various fields of multivariate analysis. Chap­ ter 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 in­ vestigated; 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.


Law of large numbers Operations Research Variance data analysis econometrics lift zonoids measure multivariate analysis multivariate dispersion probability measure statistics zonoids

Authors and affiliations

  • Karl Mosler
    • 1
  1. 1.Seminar für Wirtschafts- und SozialstatistikUniversität KölnKölnGermany

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York, Inc. 2002
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-95412-7
  • Online ISBN 978-1-4613-0045-8
  • Series Print ISSN 0930-0325
  • Buy this book on publisher's site