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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
The book introduces a new representation of probability measures — the lift zonoid representation — and demonstrates its usefulness in multivariate analysis. A measure on the Euclidean d-space is represented by a convex set in (d + 1)-space, its lift zonoid. This yields an embedding of the d-variate measures into the space of symmetric convex compacts in ℝd+1. The embedding map is positive homogeneous, additive, and continuous. It has many applications in data analysis as well as in inference and in the comparison of random vectors.
Karl Mosler

2. Zonoids and lift zonoids

Abstract
This chapter contains the general theory of lift zonoids. The lift zonoid of a given measure on ℝ d is the zonoid of a related measure on ℝd+1. Therefore, first the zonoid of a measure is investigated in Sections 2.1.1 and 2.1.2. Three definitions are provided and their equivalence is proved. The zonoid of a measure whose first moment exists is a convex compact in ℝ d ; it is centrally symmetric and contains the origin. If the measure has finite support the zonoid comes out to be a polytope and is named the zonotope of the measure.
Karl Mosler

3. Central regions

Abstract
An important task of data analysis consists in identifying a part of the data which represents the typical features of the data generating process, while the remaining data points are seen as less typical or less probable regarding their hypothesized distribution. Often the data points are assumed to vary around a center and a central region is sought which includes the center and reflects the location and general shape of the data. Such a central region can also separate one or more outlying data points from the main body of the distribution. The border of a central region, called contour set, serves as a multivariate quantile.
Karl Mosler

4. Data depth

Abstract
A data depth is a function that indicates, in some sense, how deep a point is located with respect to a given data cloud (or to a given probability distribution) in d-space. The depth defines a center of the cloud, that is the set of deepest points, and measures how far away a point is located from the center. Various notions of data depth can be employed in procedures of multivariate data analysis, such as cluster analysis and the detection of outlying data. In multivariate statistical inference they are also used to construct rank tests for homogeneity against scale and location alternatives.
Karl Mosler

5. Inference based on data depth by Rainer Dyckerhoff

Abstract
Tests which are based on the rank statistic play an important role in univariate nonparametric statistics. The rank statistic of a sample assigns to each sample variable its position in the order statistic. If one tries to generalize rank tests to the multivariate case, one is faced with the problem of defining a multivariate order statistic. Because of the absence of a natural linear order on ℝ d it is not clear how to define a multivariate order or rank statistic in a meaningful way.
Rainer Dykerhoff

6. Depth of hyperplanes

Abstract
So far, notions of data depths have been discussed that measure how deep a single point is located in a distribution. In many applications, particularly two-sample problems, a statistic is needed that measures how deep a set of points is located in a distribution. For this, the notion of data depth of a set or, more precisely, a data matrix has to be developed. Like in the case of point data depth, many notions are possible and, depending on the application, useful.
Karl Mosler

7. Volume statistics

Abstract
The volume of the lift zonoid is a parameter that reflects the dispersion of a distribution. The volume is zero if the distribution concentrates at one point, it increases if the distribution becomes more diverse. For an empirical distribution the lift zonoid volume is easily calculated; it serves as a statistic to describe the dispersion of the data.
Karl Mosler

8. Orderings and indices of dispersion

Abstract
The problem of comparing random vectors with respect to their dispersion is common to many parts of applied probability and statistics, among them estimation problems and the comparison of experiments.
Karl Mosler

9. Measuring economic disparity and concentration

Abstract
This chapter treats orders of multivariate dispersion which are particularly designed for economic applications, namely to measure economic inequality and industrial concentration.
Karl Mosler

Backmatter

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