Multivariate Dispersion, Central Regions, and Depth

The Lift Zonoid Approach
Softcover Nachdruck of the original 1. Auflage 2002
ISBN: 978-0-387-95412-7
Verlag: Springer

Buch, Englisch, Band 165, 292 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 476 g

Reihe: Lecture Notes in Statistics

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.

Zielgruppe

Research

Autoren/Hrsg.

Mosler, Karl

Fachgebiete

• Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik
• Mathematik | Informatik Mathematik Stochastik Elementare Stochastik
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
• Mathematik | Informatik Mathematik Mathematische Analysis
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

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.