Multivariate Dispersion, Central Regions, and Depth
The Lift Zonoid Approach
(Sprache: Englisch)
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...
Leider schon ausverkauft
versandkostenfrei
Buch (Kartoniert)
106.99 €
53 DeutschlandCard Punkte
sammeln
- Lastschrift, Kreditkarte, Paypal, Rechnung
- Kostenlose Rücksendung
- Ratenzahlung möglich
Produktdetails
Produktinformationen zu „Multivariate Dispersion, Central Regions, and Depth “
Klappentext zu „Multivariate Dispersion, Central Regions, and Depth “
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.
The lift zonoid approach is based on a new representation of probability measures: a d-variate probability measure is represented by a convex set, its lift zonoid. First, lift zonoids are useful in data analysis to describe an empiricaldistribution by central (so- called trimmed) regions. They give rise to a concept of data depth related to the mean which is also useful in nonparametric tests for location and scale. Second, for comparing random vectors, the set inclusion of lift zonoids defines a stochastic order that reflects the dispersion of random vectors. This has many applications to stochastic comparison problems in economics and other fields. This monograph ves the first account in book form of the theory of lift zonoids and demonstrates its usefulness in multivariate analysis. Chapter 1 offers the reader an informal introduction to basic ideas, Chapter 2 presents a comprehensive investigation into the theory. The remaining seven chapters treat various applications of the lift zonoid approach and may be separately studied. Readers are assumed to have a firm grounding in probability at the graduate level. Karl Mosler is Professor of Statistics and Econometrics at the University of Cologne. He is Editor of the Allgemeines Statistisches Archive, Journal of the German Statistical Society, and has authored numerous research articles and four books (all with Springer-Verlag) in statistics and operations research.
Inhaltsverzeichnis zu „Multivariate Dispersion, Central Regions, and Depth “
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
... mehr
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.
... weniger
Autoren-Porträt von Karl Mosler
Karl Mosler hat seine statistische und mathematische Ausbildung in Heidelberg und München erhalten. Er hat Statistik und Operations Research an wirtschaftswissenschaftlichen Fakultäten, u.a. in Hamburg und Frankfurt/O. gelehrt. Seit 1995 ist er Professor für Statistik und Ökonometrie an der Universität zu Köln.
Bibliographische Angaben
- Autor: Karl Mosler
- 2002, Softcover reprint of the original 1st ed. 2002, XII, 292 Seiten, Maße: 15,5 x 23,5 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 0387954120
- ISBN-13: 9780387954127
- Erscheinungsdatum: 10.07.2002
Sprache:
Englisch
Kommentar zu "Multivariate Dispersion, Central Regions, and Depth"
0 Gebrauchte Artikel zu „Multivariate Dispersion, Central Regions, and Depth“
| Zustand | Preis | Porto | Zahlung | Verkäufer | Rating |
|---|
Schreiben Sie einen Kommentar zu "Multivariate Dispersion, Central Regions, and Depth".
Kommentar verfassen