Multivariate Dispersion, Central Regions, and Depth
The Lift Zonoid Approach
Published on 10. July 2002
Book
Paperback/Softback
XII, 292 pages
978-0-387-95412-7 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 2002
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
XII, 292 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 17 mm
Weight
476 gr
ISBN-13
978-0-387-95412-7 (9780387954127)
DOI
10.1007/978-1-4613-0045-8
Schweitzer Classification
Person
Karl Mosler hat Mathematik und Statistik an den Universitäten Heidelberg und München studiert sowie an der Technischen Universität München promoviert. Er lehrte Statistik und Operations Research, u.a. in Hamburg und Frankfurt/Oder. 1995 erhielt er die Professur für Statistik und Ökonometrie an der Universität zu Köln.
Rainer Dyckerhoff hat an der Technischen Universität Darmstadt Mathematik studiert. Nach seiner Promotion zum Dr. rer. pol. an der Helmut-Schmidt-Universität Hamburg wechselte er 1995 an die Universität zu Köln, wo er, seit 2013 als apl. Professor, Statistik und Mathematik lehrt.
Christoph Scheicher hat an der Universität Siegen Wirtschaftsmathematik studiert. 2009 promovierte er an der Universität zu Köln und lehrt dort seit 2007 u.a. Mathematische Methoden für Ökonomen.
Content
Preface.- 1 Introduction.- 1.4 Examples of lift zonoids.- 1.5 Representing distributions by convex compacts.- 1.6 Ordering distributions.- 1.7 Central regions and data depth.- 1.8 Statistical inference.- 2 Zonoids and lift zonoids.- 2.1 Zonotopes and zonoids.- 2.2 Lift zonoid of a measure.- 2.3 Embedding into convex compacts.- 2.4 Continuity and approximation.- 2.5 Limit theorems.- 2.6 Representation of measures by a functional.- 2.7 Notes.- 3 Central regions.- 3.1 Zonoid trimmed regions.- 3.2 Properties.- 3.3 Univariate central regions.- 3.4 Examples of zonoid trimmed regions.- 3.5 Notions of central regions.- 3.6 Continuity and law of large numbers.- 3.7 Further properties.- 3.8 Trimming of empirical measures.- 3.9 Computation of zonoid trimmed regions.- 3.10 Notes.- 4 Data depth.- 4.1 Zonoid depth.- 4.2 Properties of the zonoid depth.- 4.3 Different notions of data depth.- 4.4 Combination invariance.- 4.5 Computation of the zonoid depth.- 4.6 Notes.- 5 Inference based on data depth (by Rainer Dyckerhoff).- 5.1 General notion of data depth.- 5.2 Two-sample depth test for scale.- 5.3 Two-sample rank test for location and scale.- 5.4 Classical two-sample tests.- 5.5 A new Wilcoxon distance test.- 5.6 Power comparison.- 5.7 Notes.- 6 Depth of hyperlanes.- 6.1 Depth of a hyperlane and MHD of a sample.- 6.2 Properties of MHD and majority depth.- 6.3 Combinatorial invariance.- 6.4 measuring combinatorial dispersion.- 6.5 MHD statistics.- 6.6 Significance tests and their power.- 6.7 Notes.- 7 Depth of hyperlanes.- 6.1 Depth of a hyperplane and MHD of a sample.- 6.2 Properties of MHD and majority depth.- 6.3 Combinatorial invariance.- 6.4 Measuring combinatorial dispersion.- 6.5 MHD statistics.- 6.6 Significance tests and their power.- 6.7 Notes.- 8 Orderings and indices of dispersion.- 8.1 Lift zonoid order.- 8.2 order of marginals and independence.- 8.3 Order of convolutions.- 8.4 Lift zonoid order vs. convex order.- 8.5 Volume inequalities and random determinants.- 8.6 Increasing, scaled, and centered orders.- 8.7 Properties of dispersion orders.- 8.8 Multivariate indices of dispersion.- 8.9 Notes.- 9 Economic disparity and concentration.- 9.1 Measuring economic inequality.- 9.2 Inverse Lorenz function (ILF).- 9.3 Price Lorenz order.- 9.4 Majorizations of absolute endowments.- 9.5 Other inequality orderings.- 9.6 Measuring industrial concentration.- 9.7 Multivariate concentration function.- 9.8 Multivariate concentration indices.- 9.9 Notes.- Appendix A: Basic notions.- Appendix B: Lift zonoids of bivariate normals.