Buch, Englisch, 292 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 476 g
Reihe: Lecture Notes in Statistics
The Lift Zonoid Approach
Buch, Englisch, 292 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 476 g
Reihe: Lecture Notes in Statistics
ISBN: 978-0-387-95412-7
Verlag: Springer
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.
Fachgebiete
- Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik
- Mathematik | Informatik Mathematik Mathematische Analysis
- Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
- Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
- Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
- Mathematik | Informatik Mathematik Stochastik Elementare Stochastik
- Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
Weitere Infos & Material
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.
