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     Uni Köln > WiSo-Fakultät > Seminar für Wirtschafts- und Sozialstatistik > Forschung > Prof. Mosler > Multivariate Dispersion, ...

Multivariate Despersion, Central Regions and Depth

 

Abstract

Preface

Contents

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