Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach

Front Cover
Springer Science & Business Media, 2002 M07 10 - 291 pages
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.

From inside the book

What people are saying - Write a review

We haven't found any reviews in the usual places.

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
CVIII
285
Copyright

Other editions - View all

Common terms and phrases

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.

Bibliographic information