Multivariate Nonparametric Methods with R (eBook)

Notation 11
1 Introduction 12
2 Multivariate location and scatter models 16
2.1 Construction of the multivariate models 16
2.2 Multivariate elliptical distributions 19
2.3 Other distribution families 22
3 Location and scatter functionals and sample statistics 25
3.1 Location and scatter functionals 25
3.2 Location and scatter statistics 28
3.3 First and second moments of location and scatter statistics 29
3.4 Breakdown point 33
3.5 Influence function and asymptotics 34
3.6 Other uses of location and scatter statistics 35
4 Multivariate signs and ranks 38
4.1 The use of score functions 38
4.2 Univariate signs and ranks 41
4.3 Multivariate spatial signs and ranks 44
4.4 Sign and rank covariance matrices 50
4.5 Other approaches 53
5 One-sample problem: Hotelling's T2-test 56
5.1 Example 56
5.2 General strategy for estimation and testing 58
5.3 Hotelling's T2-test 61
6 One-sample problem: Spatial sign test and spatial median 67
6.1 Multivariate spatial sign test 67
6.1.1 Preliminaries 67
6.1.2 The test outer standardization 68
6.1.3 The test with inner standardization 73
6.1.4 Other sign-based approaches for testing problem 76
6.2 Multivariate spatial median 78
6.2.1 The regular spatial median 78
6.2.2 The estimate with inner standardization 83
6.2.3 Other multivariate medians 88
7 One-sample problem: Spatial signed-rank test and Hodges-Lehmann estimate 90
7.1 Multivariate spatial signed-rank test 90
7.2 Multivariate spatial Hodges-Lehmann estimate 96
7.3 Other approaches 99
8 One-sample problem: Comparisons of tests and estimates 102
8.1 Asymptotic relative efficiencies 102
8.2 Finite sample comparisons 105
9 One-sample problem: Inference for shape 113
9.1 The estimation and testing problem 113
9.2 Important matrix tools 115
9.3 The general strategy for estimation and testing 116
9.4 Test and estimate based on UCOV 118
9.5 Test and estimates based on TCOV 120
9.6 Tests and estimates based on RCOV 121
9.7 Limiting efficiencies 123
9.8 Examples 124
9.9 Principal component analysis based on spatial signs and ranks 129
9.10 Other approaches 135
10 Multivariate tests of independence 136
10.1 The problem and a general strategy 136
10.2 Wilks' and Pillai's tests 137
10.3 Tests based on spatial signs and ranks 138
10.4 Efficiency comparisons 141
10.5 A real data example 143
10.6 Canonical correlation analysis 146
10.7 Other approaches 147
11 Several-sample location problem 149
11.1 A general strategy for testing and estimation 149
11.2 Hotelling's T2 and MANOVA 151
11.3 The test based on spatial signs 157
11.4 The tests based on spatial ranks 161
11.5 Estimation of the treatment effects 164
11.6 An example: Egyptian skulls from three epochs. 167
11.7 References and other approaches 172
12 Randomized blocks 175
12.1 The problem and the test statistics 175
12.2 Limiting distributions and efficiency 179
12.3 Treatment effect estimates 180
12.4 Affine invariant tests and affine equivariant estimates 183
12.5 Examples and final remarks 183
12.6 Other approaches 185
13 Multivariate linear regression 187
13.1 General strategy 187
13.2 Multivariate linear L2 regression 191
13.3 L1 regression based on spatial signs 193
13.4 L1 regression based on spatial ranks 197
13.5 An example 200
13.6 Other approaches 204
14 Analysis of cluster-correlated data 205
14.1 Introduction 205
14.2 One-sample case 206
14.2.1 Notation and assumptions 206
14.2.2 Tests and estimates 207
14.2.3 Tests and estimates, weighted versions 208
14.3 Two samples: Weighted spatial rank test 210
14.4 References and other approaches 212
A Some vector and matrix algebra 213
B Asymptotical results for methods based on spatial signs 218
B.1 Some auxiliary results 218
B.2 Basic limit theorems 219
B.3 Notation and assumptions 220
B.4 Limiting results for spatial median 221
B.5 Limiting results for the multivariate regression estimate 222
Preface 6
References 
224 
Index 234