Oja Multivariate Nonparametric Methods with R

An approach based on spatial signs and ranks
1. Auflage 2010
ISBN: 978-1-4419-0468-3
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 234 Seiten

Reihe: Lecture Notes in Statistics

This book offers a new, fairly efficient, and robust alternative to analyzing multivariate data. The analysis of data based on multivariate spatial signs and ranks proceeds very much as does a traditional multivariate analysis relying on the assumption of multivariate normality, the regular L2 norm is just replaced by different L1 norms, observation vectors are replaced by spatial signs and ranks, and so on. A unified methodology starting with the simple one-sample multivariate location problem and proceeding to the general multivariate multiple linear regression case is presented. Companion estimates and tests for scatter matrices are considered as well. The R package MNM is available for computation of the procedures. This monograph provides an up-to-date overview of the theory of multivariate nonparametric methods based on spatial signs and ranks. The classical book by Puri and Sen (1971) uses marginal signs and ranks and different type of L1 norm. The book may serve as a textbook and a general reference for the latest developments in the area. Readers are assumed to have a good knowledge of basic statistical theory as well as matrix theory. TOC:Introduction.- Multivariate location and scatter models.- Location and scatter functionals and sample statistics.- Multivariate signs and ranks.- One-sample problems: Hotelling`s T2-test.- One-sample problem: Spatial sign test and spatial median.- One-sample problem: Spatial signed-rank test and Hodges-Lehmann estimate.- One-sample problem: Comparisons of tests and estimates.- One-sample problem: Inference for shape.- Multivariate tests of independence.- Several-sample location problem.- Randomized blocks.- Multivariate linear regression.- Analysis of cluster-correlated data.

Autoren/Hrsg.

Oja, Hannu

Inhaltsverzeichnis

1;Notation;11
2;1 Introduction;12
3;2 Multivariate location and scatter models;16
3.1;2.1 Construction of the multivariate models;16
3.2;2.2 Multivariate elliptical distributions;19
3.3;2.3 Other distribution families;22
4;3 Location and scatter functionals and sample statistics;25
4.1;3.1 Location and scatter functionals;25
4.2;3.2 Location and scatter statistics;28
4.3;3.3 First and second moments of location and scatter statistics;29
4.4;3.4 Breakdown point;33
4.5;3.5 Influence function and asymptotics;34
4.6;3.6 Other uses of location and scatter statistics;35
5;4 Multivariate signs and ranks;38
5.1;4.1 The use of score functions;38
5.2;4.2 Univariate signs and ranks;41
5.3;4.3 Multivariate spatial signs and ranks;44
5.4;4.4 Sign and rank covariance matrices;50
5.5;4.5 Other approaches;53
6;5 One-sample problem: Hotelling's T2-test;56
6.1;5.1 Example;56
6.2;5.2 General strategy for estimation and testing;58
6.3;5.3 Hotelling's T2-test;61
7;6 One-sample problem: Spatial sign test and spatial median;67
7.1;6.1 Multivariate spatial sign test;67
7.1.1;6.1.1 Preliminaries;67
7.1.2;6.1.2 The test outer standardization;68
7.1.3;6.1.3 The test with inner standardization;73
7.1.4;6.1.4 Other sign-based approaches for testing problem;76
7.2;6.2 Multivariate spatial median;78
7.2.1;6.2.1 The regular spatial median;78
7.2.2;6.2.2 The estimate with inner standardization;83
7.2.3;6.2.3 Other multivariate medians;88
8;7 One-sample problem: Spatial signed-rank test and Hodges-Lehmann estimate;90
8.1;7.1 Multivariate spatial signed-rank test;90
8.2;7.2 Multivariate spatial Hodges-Lehmann estimate;96
8.3;7.3 Other approaches;99
9;8 One-sample problem: Comparisons of tests and estimates;102
9.1;8.1 Asymptotic relative efficiencies;102
9.2;8.2 Finite sample comparisons;105
10;9 One-sample problem: Inference for shape;113
10.1;9.1 The estimation and testing problem;113
10.2;9.2 Important matrix tools;115
10.3;9.3 The general strategy for estimation and testing;116
10.4;9.4 Test and estimate based on UCOV;118
10.5;9.5 Test and estimates based on TCOV;120
10.6;9.6 Tests and estimates based on RCOV;121
10.7;9.7 Limiting efficiencies;123
10.8;9.8 Examples;124
10.9;9.9 Principal component analysis based on spatial signs and ranks;129
10.10;9.10 Other approaches;135
11;10 Multivariate tests of independence;136
11.1;10.1 The problem and a general strategy;136
11.2;10.2 Wilks' and Pillai's tests;137
11.3;10.3 Tests based on spatial signs and ranks;138
11.4;10.4 Efficiency comparisons;141
11.5;10.5 A real data example;143
11.6;10.6 Canonical correlation analysis;146
11.7;10.7 Other approaches;147
12;11 Several-sample location problem;149
12.1;11.1 A general strategy for testing and estimation;149
12.2;11.2 Hotelling's T2 and MANOVA;151
12.3;11.3 The test based on spatial signs;157
12.4;11.4 The tests based on spatial ranks;161
12.5;11.5 Estimation of the treatment effects;164
12.6;11.6 An example: Egyptian skulls from three epochs.;167
12.7;11.7 References and other approaches;172
13;12 Randomized blocks;175
13.1;12.1 The problem and the test statistics;175
13.2;12.2 Limiting distributions and efficiency;179
13.3;12.3 Treatment effect estimates;180
13.4;12.4 Affine invariant tests and affine equivariant estimates;183
13.5;12.5 Examples and final remarks ;183
13.6;12.6 Other approaches;185
14;13 Multivariate linear regression;187
14.1;13.1 General strategy;187
14.2;13.2 Multivariate linear L2 regression ;191
14.3;13.3 L1 regression based on spatial signs;193
14.4;13.4 L1 regression based on spatial ranks;197
14.5;13.5 An example;200
14.6;13.6 Other approaches;204
15;14 Analysis of cluster-correlated data;205
15.1;14.1 Introduction;205
15.2;14.2 One-sample case;206
15.2.1;14.2.1 Notation and assumptions;206
15.2.2;14.2.2 Tests and estimates;207
15.2.3;14.2.3 Tests and estimates, weighted versions;208
15.3;14.3 Two samples: Weighted spatial rank test;210
15.4;14.4 References and other approaches;212
16;A Some vector and matrix algebra;213
17;B Asymptotical results for methods based on spatial signs;218
17.1;B.1 Some auxiliary results;218
17.2;B.2 Basic limit theorems;219
17.3;B.3 Notation and assumptions;220
17.4;B.4 Limiting results for spatial median;221
17.5;B.5 Limiting results for the multivariate regression estimate;222
18;Preface;6
19;References
;224
20;Index;234