# Multivariate Dispersion, Central Regions, and Depth

## The Lift Zonoid Approach

• Karl Mosler
Book

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

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. Rainer Dykerhoff
Pages 133-163
7. Karl Mosler
Pages 165-179
8. Karl Mosler
Pages 181-206
9. Karl Mosler
Pages 207-228
10. Karl Mosler
Pages 229-258
11. Back Matter
Pages 259-295

### Introduction

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.

### Keywords

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 https://doi.org/10.1007/978-1-4613-0045-8
• Copyright Information Springer-Verlag New York, Inc. 2002
• Publisher Name Springer, New York, NY
• eBook Packages
• 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
Industry Sectors
Pharma