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＜電子ブック＞
Multivariate Dispersion, Central Regions, and Depth : The Lift Zonoid Approach / by Karl Mosler
(Lecture Notes in Statistics. ISSN:21977186 ; 165)

版	1st ed. 2002.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	2002
大きさ	XII, 292 p : online resource
著者標目	*Mosler, Karl author SpringerLink (Online service)
件　名	LCSH:Mathematical analysis LCSH:Statistics  FREE:Analysis FREE:Statistics in Business, Management, Economics, Finance, Insurance FREE:Statistical Theory and Methods
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-1-4613-0045-8

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