Preface
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.
Acknowledgments
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
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