Multivariate Analysis - Charles C. Taylor

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Résumé :
Multivariate Analysis

Comprehensive Reference Work on Multivariate Analysis and its Applications

The first edition of this book, by Mardia, Kent and Bibby, has been used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments.

A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of:

• Basic properties of random vectors, copulas, normal distribution theory, and estimation
• Hypothesis testing, multivariate regression, and analysis of variance
• Principal component analysis, factor analysis, and canonical correlation analysis
• Discriminant analysis, cluster analysis, and multidimensional scaling
• New advances and techniques, including supervised and unsupervised statistical learning, graphical models and regularization methods for high-dimensional data

Although primarily designed as a textbook for final year undergraduates and postgraduate students in mathematics and statistics, the book will also be of interest to research workers and applied scientists....

Biographie:

Kanti V. Mardia OBE is a Senior Research Professor in the Department of Statistics at the University of Leeds, Leverhulme Emeritus Fellow, and Visiting Professor in the Department of Statistics, University of Oxford.

John T. Kent and Charles C. Taylor are both Professors in the Department of Statistics, University of Leeds....

Sommaire:

Epigraph xvii

Preface to the Second Edition xix

Preface to the First Edition xxi

Acknowledgments from First Edition xxv

Notation, Abbreviations, and Key Ideas xxvii

1 Introduction 1

1.1 Objects and Variables 1

1.2 Some Multivariate Problems and Techniques 1

1.3 The Data Matrix 7

1.4 Summary Statistics 8

1.5 Linear Combinations 12

1.6 Geometrical Ideas 14

1.7 Graphical Representation 15

1.8 Measures of Multivariate Skewness and Kurtosis 20

Exercises and Complements 22

2 Basic Properties of Random Vectors 25

Introduction 25

2.1 Cumulative Distribution Functions and Probability Density Functions 25

2.2 Population Moments 27

2.3 Characteristic Functions 31

2.4 Transformations 32

2.5 The Multivariate Normal Distribution 34

2.6 Random Samples 41

2.7 Limit Theorems 42

Exercises and Complements 44

3 Nonnormal Distributions 49

3.1 Introduction 49

3.2 Some Multivariate Generalizations of Univariate Distributions 49

3.3 Families of Distributions 52

3.4 Insights into Skewness and Kurtosis 57

3.5 Copulas 60

Exercises and Complements 65

4 Normal Distribution Theory 71

4.1 Introduction and Characterization 71

4.2 Linear Forms 73

4.3 Transformations of Normal Data Matrices 75

4.4 The Wishart Distribution 77

4.5 The Hotelling T² Distribution 83

4.6 Mahalanobis Distance 85

4.7 Statistics Based on the Wishart Distribution 88

4.8 Other Distributions Related to the Multivariate Normal 92

Exercises and Complements 93

5 Estimation 101

Introduction 101

5.1 Likelihood and Sufficiency 101

5.2 Maximum-likelihood Estimation 106

5.3 Robust Estimation of Location and Dispersion for Multivariate Distributions 112

5.4 Bayesian Inference 117

Exercises and Complements 119

6 Hypothesis Testing 125

6.1 Introduction 125

6.2 The Techniques Introduced 127

6.3 The Techniques Further Illustrated 134

6.4 Simultaneous Confidence Intervals 142

6.5 The Behrens-Fisher Problem 144

6.6 Multivariate Hypothesis Testing: Some General Points 145

6.7 Nonnormal Data 146

6.8 Mardia's Nonparametric Test for the Bivariate Two-sample Problem 149

Exercises and Complements 151

7 Multivariate Regression Analysis 159

7.1 Introduction 159

7.2 Maximum-likelihood Estimation 160

7.3 The General Linear Hypothesis 162

7.4 Design Matrices of Degenerate Rank 165

7.5 Multiple Correlation 167

7.6 Least-squares Estimation 171

7.7 Discarding of Variables 174

Exercises and Complements 178

8 Graphical Models 183

8.1 Introduction 183

8.2 Graphs and Conditional Independence 184

8.3 Gaussian Graphical Models 188

8.4 Log-linear Graphical Models 195

8.5 Directed and Mixed Graphs 202

Exercises and Complements 204

9 Principal Component Analysis 207

9.1 Introduction 207

9.2 Definition and Properties of Principal Components 207

9.3 Sampling Properties of Principal Components 221

9.4 Testing Hypotheses About Principal Components 227

9.5 Correspondence Analysis 230

9.6 Allometry - Measurement of Size and Shape 237

9.7 Discarding of Variables 240

9.8 Principal Component Regression 241

9.9 Projection Pursuit and Independent Component Analysis 244

9.10 PCA in High Dimensions 247

Exercises and Complements 249

10 Factor Analysis 259

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