## Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid ApproachThis 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. |

### From inside the book

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### Contents

II | 1 |

III | 2 |

IV | 4 |

V | 9 |

VI | 14 |

VII | 16 |

VIII | 19 |

IX | 22 |

LV | 134 |

LVI | 136 |

LVII | 139 |

LVIII | 141 |

LX | 142 |

LXI | 144 |

LXII | 145 |

LXIII | 147 |

X | 25 |

XI | 27 |

XIII | 30 |

XIV | 32 |

XV | 34 |

XVI | 35 |

XVII | 38 |

XVIII | 40 |

XX | 43 |

XXI | 48 |

XXII | 49 |

XXIII | 50 |

XXIV | 51 |

XXV | 52 |

XXVI | 55 |

XXVII | 58 |

XXVIII | 59 |

XXIX | 65 |

XXX | 66 |

XXXI | 68 |

XXXII | 71 |

XXXIII | 74 |

XXXIV | 77 |

XXXVI | 79 |

XXXVII | 81 |

XXXVIII | 84 |

XXXIX | 85 |

XL | 88 |

XLI | 93 |

XLII | 96 |

XLIII | 97 |

XLIV | 100 |

XLV | 102 |

XLVI | 103 |

XLVII | 105 |

XLVIII | 108 |

XLIX | 111 |

L | 116 |

LI | 123 |

LII | 127 |

LIII | 130 |

LIV | 133 |

LXIV | 149 |

LXV | 160 |

LXVI | 162 |

LXVIII | 163 |

LXIX | 165 |

LXX | 168 |

LXXI | 170 |

LXXII | 171 |

LXXIV | 176 |

LXXV | 178 |

LXXVI | 179 |

LXXVII | 183 |

LXXVIII | 185 |

LXXIX | 188 |

LXXX | 194 |

LXXXI | 195 |

LXXXII | 202 |

LXXXIII | 204 |

LXXXV | 205 |

LXXXVI | 210 |

LXXXVII | 211 |

LXXXVIII | 213 |

LXXXIX | 216 |

XCI | 219 |

XCII | 221 |

XCIII | 225 |

XCIV | 226 |

XCVI | 227 |

XCVII | 229 |

XCVIII | 235 |

XCIX | 239 |

C | 242 |

CI | 245 |

CII | 249 |

CIII | 252 |

CIV | 254 |

CV | 256 |

CVI | 262 |

CVII | 271 |

285 | |

### Other editions - View all

Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach Karl Mosler Limited preview - 2012 |

### Common terms and phrases

a-trimmed affine equivariant affine invariant affine transformation algorithm Cauchy Cauchy distribution central regions combinatorial invariant computation consider convergence convex hull convex order convex sets Corollary covariance d-variate Da(n data depth defined definition denote dependence order depth tests dimension dispersion distribution function elliptical distributions empirical distribution empirical measure equals equivariant Example extreme points FIGURE finite follows Gini index Gini mean difference given graph halfspace depth Hausdorff distance holds hyperplane Koshevoy and Mosler L-continuous large numbers law of large lift zonoid order lift zonotope linear Lorenz curve Lorenz zonoid Mahalanobis depth matrix median minimal spanning tree normal distribution Oja depth price Lorenz order probability distributions probability measure Proof properties Proposition Puri-Sen test Radon partition random vectors rank respect Rrf+i scale Section sequence stochastic order support function symmetric test statistic Theorem transformation uniformly integrable uniquely univariate zonoid depth zonoid trimmed regions zonotope

### Popular passages

Page 275 - Liu, RY, Parelius, JM and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion).

Page 281 - Structural properties and convergence results for contours of sample statistical depth functions.

Page 275 - Methods of Measuring the Concentration of Wealth. Publication of the American Statistical Association, New Series, 70, 1905, 20921.

Page 281 - ZUO. Y. and SERFLING. R. (2000a). General notions of statistical depth function. Annals of Statistics 28, 461-482. ZUO. Y. and SERFLING. R. (2000b). Nonparametric notions of multivariate "scatter measure'' and "more scattered" based on statistical depth functions.

Page 281 - On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. Journal of Statistical Plannmg and Inference 84, 55-79.

Page 275 - Masse. JC and Theodorescu. R. (1994). Halfplane trimming for bivariate distributions, Journal of Multivariate Analysis, 48, 188-202.