Giri / Birnbaum / Lukacs Multivariate Statistical Inference

1. Auflage 2014
ISBN: 978-1-4832-6333-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 336 Seiten, Web PDF

Multivariate Statistical Inference is a 10-chapter text that covers the theoretical and applied aspects of multivariate analysis, specifically the multivariate normal distribution using the invariance approach. Chapter I contains some special results regarding characteristic roots and vectors, and partitioned submatrices of real and complex matrices, as well as some special theorems on real and complex matrices useful in multivariate analysis. Chapter II deals with the theory of groups and related results that are useful for the development of invariant statistical test procedures, including the Jacobians of some specific transformations that are useful for deriving multivariate sampling distributions. Chapter III is devoted to basic notions of multivariate distributions and the principle of invariance in statistical testing of hypotheses. Chapters IV and V deal with the study of the real multivariate normal distribution through the probability density function and through a simple characterization and the maximum likelihood estimators of the parameters of the multivariate normal distribution and their optimum properties. Chapter VI tackles a systematic derivation of basic multivariate sampling distributions for the real case, while Chapter VII explores the tests and confidence regions of mean vectors of multivariate normal populations with known and unknown covariance matrices and their optimum properties. Chapter VIII is devoted to a systematic derivation of tests concerning covariance matrices and mean vectors of multivariate normal populations and to the study of their optimum properties. Chapters IX and X look into a treatment of discriminant analysis and the different covariance models and their analysis for the multivariate normal distribution. These chapters also deal with the principal components, factor models, canonical correlations, and time series. This book will prove useful to statisticians, mathematicians, and advance mathematics students.

Autoren/Hrsg.

Giri, Narayan C.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Multivariate Statistical Inference;6
3;Copyright Page;7
4;Table of Contents;8
5;Dedication;7
6;Preface;12
7;Acknowledgments;16
8;CHAPTER I. Vector and Matrix Algebra;18
8.1;1.0 INTRODUCTION;18
8.2;1.1 VECTORS;18
8.3;1.2 MATRICES;20
8.4;1.3 RANK AND TRACE OF A MATRIX;23
8.5;1.4 QUADRATIC FORMS AND POSITIVE DEFINITE MATRIX;24
8.6;1.5 CHARACTERISTIC ROOTS AND VECTORS;25
8.7;1.6 PARTITIONED MATRIX;30
8.8;1.7 SOME SPECIAL THEOREMS;33
8.9;1.8 COMPLEX MATRICES;35
8.10;EXERCISES;35
8.11;REFERENCES;37
9;CHAPTER II. Groups and Jacobian of Some Transformations;38
9.1;2.0 INTRODUCTION;38
9.2;2.1 GROUPS;38
9.3;2.2 SOME EXAMPLES OF GROUPS;39
9.4;2.3 NORMAL SUBGROUPS, QUOTIENT GROUP, HOMOMORPHISM, ISOMORPHISM, DIRECT PRODUCT;40
9.5;2.4 JACOBIAN OF SOME TRANSFORMATIONS;42
9.6;FURTHER READING;45
10;CHAPTER III. Notions of Multivariate Distributions and Invariance in Statistical Inference;46
10.1;3.0 INTRODUCTION;46
10.2;3.1 MULTIVARIATE DISTRIBUTIONS;46
10.3;3.2 INVARIANCE IN STATISTICAL TESTING OF HYPOTHESES;49
10.4;3.3 SUFFICIENCY AND INVARIANCE;59
10.5;3.4 UNBIASEDNESS AND INVARIANCE;59
10.6;3.5 INVARIANCE AND OPTIMUM TESTS;60
10.7;3.6 MOST STRINGENT TESTS AND INVARIANCE;61
10.8;EXERCISES;62
10.9;REFERENCES;64
11;CHAPTER IV. Multivariate Normal Distribution, Its Properties and Characterization;66
11.1;4.0 INTRODUCTION;66
11.2;4.1 MULTIVARIATE NORMAL DISTRIBUTION (CLASSICAL APPROACH);67
11.3;4.2 SOME CHARACTERIZATIONS OF THE NORMAL DISTRIBUTION;74
11.4;4.3 COMPLEX MULTIVARIATE NORMAL DISTRIBUTION;75
11.5;4.4 CONCENTRATION ELLIPSOID AND AXES;76
11.6;4.5 SOME EXAMPLES;78
11.7;4.6 REGRESSION, MULTIPLE AND PARTIAL CORRELATION;80
11.8;EXERCISES;83
11.9;REFERENCES;88
12;CHAPTER V. Estimators of Parameters and Their Functions in a Multivariate Normal Distribution;89
12.1;5.0 INTRODUCTION;89
12.2;5.1 MAXIMUM LIKELIHOOD ESTIMATORS OF µ, S;90
12.3;5.2 PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATORS OF µ AND S;96
12.4;5.3 BAYES, MINIMAX, AND ADMISSIBLE CHARACTERS OF THE MAXIMUM LIKELIHOOD ESTIMATOR OF (µ, S);106
12.5;EXERCISES;119
12.6;REFERENCES;122
13;CHAPTER VI. Basic Multivariate Sampling Distributions;124
13.1;6.0 INTRODUCTION;124
13.2;6.1 NONCENTRAL CHI-SQUARE, STUDENT'S t-, F-DISTRIBUTIONS;124
13.3;6.2 DISTRIBUTION OF QUADRATIC FORMS, COCHRAN'S THEOREM;126
13.4;6.3 THE WISHART DISTRIBUTION;130
13.5;6.4 TENSOR PRODUCT, PROPERTIES OF THE WISHART DISTRIBUTION;136
13.6;6.5 THE NONCENTRAL WISHART DISTRIBUTION;141
13.7;6.6 GENERALIZED VARIANCE;142
13.8;6.7 DISTRIBUTION OF THE BARTLETT DECOMPOSITION (RECTANGULAR COORDINATES);143
13.9;6.8 DISTRIBUTION OF HOTELLING'S T2 AND A RELATED DISTRIBUTION;144
13.10;6.9 DISTRIBUTION OF MULTIPLE AND PARTIAL CORRELATION COEFFICIENTS;150
13.11;EXERCISES;153
13.12;REFERENCES;158
14;CHAPTER VII. Tests of Hypotheses of Mean Vectors;161
14.1;7.0 INTRODUCTION;161
14.2;7.1 TESTS AND CONFIDENCE REGION FOR MEAN VECTORS WITH KNOWN COVARIANCE MATRICES;162
14.3;7.2 TESTS AND CONFIDENCE REGION FOR MEAN VECTORS WITH UNKNOWN COVARIANCE MATRICES;164
14.4;7.3 TESTS OF HYPOTHESES CONCERNING SUBVECTORS OF µ;189
14.5;EXERCISES;192
14.6;REFERENCES;194
15;CHAPTER VIII. Tests Concerning Covariance Matrices and Mean Vectors;196
15.1;8.0 INTRODUCTION;196
15.2;8.1 HYPOTHESIS THAT A COVARIANCE MATRIX IS EQUAL TO A GIVEN MATRIX;197
15.3;8.2 THE SPHERICITY TEST;203
15.4;8.3 TESTS OF INDEPENDENCE AND THE R2-TEST;207
15.5;8.4 MULTIVARIATE GENERAL LINEAR HYPOTHESIS;222
15.6;8.5 EQUALITY OF SEVERAL COVARIANCE MATRICES;240
15.7;EXERCISES;250
15.8;REFERENCES;254
16;CHAPTER IX. Discriminant Analysis;257
16.1;9.0 INTRODUCTION;257
16.2;9.1 EXAMPLES;259
16.3;9.2 FORMULATION OF THE PROBLEM OF DISCRIMINANT ANALYSIS;260
16.4;9.3 CLASSIFICATION INTO ONE OF TWO MULTIVARIATE NORMAL POPULATIONS;266
16.5;9.4 CLASSIFICATION INTO MORE THAN TWO MULTIVARIATE NORMAL POPULATIONS;287
16.6;9.5 CONCLUDING REMARKS;291
16.7;EXERCISES;292
16.8;REFERENCES;294
17;CHAPTER X. Multivariate Covariance Models;298
17.1;10.0 INTRODUCTION;298
17.2;10.1 PRINCIPAL COMPONENTS;298
17.3;10.2 FACTOR ANALYSIS;310
17.4;10.3 CANONICAL CORRELATIONS;318
17.5;10.4 TIME SERIES ANALYSIS;326
17.6;EXERCISES;327
17.7;REFERENCES;328
18;Author Index;330
19;Subject Index;333