Multivariate Nonparametric Methods with R: An approach based on spatial signs and ranks (Lecture Notes in Statistics, 199)

✍ Scribed by Hannu Oja

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 239
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Multivariate Nonparametric
Methods with R
Preface
Notation
1 Introduction
2 Multivariate location and scatter models
2.1 Construction of the multivariate models
2.2 Multivariate elliptical distributions
2.3 Other distribution families
3 Location and scatter functionals and sample statistics
3.1 Location and scatter functionals
3.2 Location and scatter statistics
3.3 First and second moments of location and scatter statistics
3.4 Breakdown point
3.5 Influence function and asymptotics
3.6 Other uses of location and scatter statistics
4 Multivariate signs and ranks
4.1 The use of score functions
4.2 Univariate signs and ranks
4.3 Multivariate spatial signs and ranks
4.4 Sign and rank covariance matrices
4.5 Other approaches
5 One-sample problem: Hotelling's T2-test
5.1 Example
5.2 General strategy for estimation and testing
5.3 Hotelling's T2-test
6 One-sample problem: Spatial sign test and spatial median
6.1 Multivariate spatial sign test
6.1.1 Preliminaries
6.1.2 The test outer standardization
6.1.3 The test with inner standardization
6.1.4 Other sign-based approaches for testing problem
6.2 Multivariate spatial median
6.2.1 The regular spatial median
6.2.2 The estimate with inner standardization
6.2.3 Other multivariate medians
7 One-sample problem: Spatial signed-rank test and Hodges-Lehmann estimate
7.1 Multivariate spatial signed-rank test
7.2 Multivariate spatial Hodges-Lehmann estimate
7.3 Other approaches
8 One-sample problem: Comparisons of tests and estimates
8.1 Asymptotic relative efficiencies
8.2 Finite sample comparisons
9 One-sample problem: Inference for shape
9.1 The estimation and testing problem
9.2 Important matrix tools
9.3 The general strategy for estimation and testing
9.4 Test and estimate based on UCOV
9.5 Test and estimates based on TCOV
9.6 Tests and estimates based on RCOV
9.7 Limiting efficiencies
9.8 Examples
9.9 Principal component analysis based on spatial signs and ranks
9.10 Other approaches
10 Multivariate tests of independence
10.1 The problem and a general strategy
10.2 Wilks' and Pillai's tests
10.3 Tests based on spatial signs and ranks
10.4 Efficiency comparisons
10.5 A real data example
10.6 Canonical correlation analysis
10.7 Other approaches
11 Several-sample location problem
11.1 A general strategy for testing and estimation
11.2 Hotelling's T2 and MANOVA
11.3 The test based on spatial signs
11.4 The tests based on spatial ranks
11.5 Estimation of the treatment effects
11.6 An example: Egyptian skulls from three epochs.
11.7 References and other approaches
12 Randomized blocks
12.1 The problem and the test statistics
12.2 Limiting distributions and efficiency
12.3 Treatment effect estimates
12.4 Affine invariant tests and affine equivariant estimates
12.5 Examples and final remarks
12.6 Other approaches
13 Multivariate linear regression
13.1 General strategy
13.2 Multivariate linear L2 regression
13.3 L1 regression based on spatial signs
13.4 L1 regression based on spatial ranks
13.5 An example
13.6 Other approaches
14 Analysis of cluster-correlated data
14.1 Introduction
14.2 One-sample case
14.2.1 Notation and assumptions
14.2.2 Tests and estimates
14.2.3 Tests and estimates, weighted versions
14.3 Two samples: Weighted spatial rank test
14.4 References and other approaches
A Some vector and matrix algebra
B Asymptotical results for methods based on spatial signs
B.1 Some auxiliary results
B.2 Basic limit theorems
B.3 Notation and assumptions
B.4 Limiting results for spatial median
B.5 Limiting results for the multivariate regression estimate
References
Index
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