UNIVERSITY OF COLOGNE

Contents

Preface	v
1 Introduction	1
1.1 The Brief	1
1.2 Representing a probability measure	2
1.3 Lift zonoids	4
1.4 Example of lift zonoids	9
1.5 Representing distributions by convex compacts	14
1.6 Ordering distributions	16
1.7 Central regions and data depth	19
1.8 Statistical inference	22
2 Zonoids and lift zonoids	25
2.1 Zonotopes and lift zonoids	27
2.1.1 Zonoid of a measure	27
2.1.2 Equivalent definitions of the zonoid of a measure	30
2.1.3 Support function of a zonoid	32
2.1.4 Zonoids as expected random segments	34
2.1.5 Volume of a zonoid	35
2.1.6 Measures with equal zonoids	38
2.2 Lift zonoid of a measure	40
2.2.1 Definition and first properties	40
2.2.2 Lift zonotope	43
2.2.3 Univariate case	43
2.3 Embedding into convex compacts	48
2.3.1 Inclusion of lift zonoids	49
2.3.2 Uniqueness of the representation	50
2.3.3 Lift zonoid metric	51
2.3.4 Linear transformations and projections	52
2.3.5 Lift zonoid of spherical and elliptical distributions	55
2.4 Continuity and approximation	58
2.4.1 Convergence of lift zonoids	59
2.4.2 Monotone approximation of measures	65
2.4.3 Volume of a lift zonid	66
2.5 Limit theorems	67
2.6 Representation of measures by a functional	70
2.6.1 Statistical representations	74
2.6.2 Lift zonoids and the empirical process	76
2.7 Notes	77
3 Central Regions	79
3.1 Zonoid trimmed regions	81
3.2 Properties	84
3.3 Univariate central regions	85
3.4 Examples of zonoid trimmed regions	88
3.5 Notions of central regions	93
3.6 Continuity and law of large numbers	96
3.7 Further properties	97
3.8 Trimming empirical measures	100
3.9 Computation of zonoid trimmed regions	102
3.10 Notes	103
4 Data Depth	105
4.1 Zonoid depth	108
4.2 Properties of the zonoid depth	111
4.3 Different notions of data depth	115
4.4 Combinatorial invariance	122
4.5 Computation of the zonoid depth	127
4.6 Notes	129
5 Inference based on data depth (by Rainer Dyckerhoff)	131
5.1 General notion of data depth	132
5.2 Two-sample depth test for scale	134
5.3 Two-sample rank test for location and scale	137
5.4 Classical two-sample tests	139
5.4.1 Box´s M Test	139
5.4.2 Friedman-Rafsky test	140
5.4.3 Hotelling´s T² test	142
5.4.4 Puri-Sen test	143
5.5 A new Wilcoxon distance test	145
5.6 Power comparison	147
5.7 Notes	161
6 Depth of hyperplanes	163
6.1 Depth of a hyperplane and MHD of a sample	164
6.2 Properties of MHD and majority depth	166
6.3 Combinatorial invariance	169
6.4 Measuring combinatorial dispersion	171
6.5 MHD statistics	172
6.6 Significance tests and their power	172
6.7 Notes	177
7 Volume statistics	179
7.1 Univariate Gini Index	180
7.2 Lift zonoid volume	184
7.3 Expected volume of a random convex hull	186
7.4 The multivariate volume-Gini index	189
7.5 Volume statistics in cluster analysis	195
7.6 Measuring dependency	196
7.7 Notes	203
8 Ordering and indices of dispersion	205
8.1 Lift zonoid order	206
8.2 Order of marginals and independence	211
8.3 Order of convolutions	212
8.4 Lift zonoid order vs. convex order	214
8.5 Volume inequalities and random determinants	217
8.6 Increasing, scaled, and centred orders	217
8.7 Properties of dispersion orders	220
8.8 Multivariate indices of dispersion	222
8.9 Notes	226
9 Economic disparity and concentration	227
9.1 Measuring economic inequality	228
9.2 Inverse Lorenz function (ILF)	230
9.3 Price Lorenz order	236
9.4 Majorizations of absolute endowments	240
9.5 Other inequality orderings	243
9.6 Measuring industrial concentration	246
9.7 Multivariate concentration function	250
9.8 Multivariate concentration indices	253
9.9 Notes	255
Appendix A: Basic Notions	257
Appendix B: Lift zonoids of bivariate normals	263
Bibliography	272
Index	286