Multivariate Dispersion, Central Regions, and Depth

1. Front Matter

   Pages i-x

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2. Introduction

   • Karl Mosler

   Pages 1-24

3. Zonoids and lift zonoids

   • Karl Mosler

   Pages 25-78

4. Central regions

   • Karl Mosler

   Pages 79-104

5. Data depth

   • Karl Mosler

   Pages 105-131

6. Inference based on data depth by Rainer Dyckerhoff

   • Rainer Dykerhoff

   Pages 133-163

7. Depth of hyperplanes

   • Karl Mosler

   Pages 165-179

8. Volume statistics

   • Karl Mosler

   Pages 181-206

9. Orderings and indices of dispersion

   • Karl Mosler

   Pages 207-228

10. Measuring economic disparity and concentration

    • Karl Mosler

    Pages 229-258

11. Back Matter

    Pages 259-295

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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.

• Law of large numbers
• Operations Research
• Variance
• data analysis
• econometrics
• lift zonoids
• measure
• multivariate analysis
• multivariate dispersion
• probability measure
• statistics
• zonoids

• Seminar für Wirtschafts- und Sozialstatistik, Universität Köln, Köln, Germany

  Karl Mosler

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