# Multivariate Statistical Inference

## 1st Edition

**Authors:**Narayan C. Giri

**Editors:**Z. W. Birnbaum E. Lukacs

**eBook ISBN:**9781483263335

**Imprint:**Academic Press

**Published Date:**1st January 1977

**Page Count:**336

## Description

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.

## Table of Contents

Preface

Acknowledgments

Chapter I Vector and Matrix Algebra

1.0 Introduction

1.1 Vectors

1.2 Matrices

1.3 Rank and Trace of a Matrix

1.4 Quadratic Forms and Positive Definite Matrix

1.5 Characteristic Roots and Vectors

1.6 Partitioned Matrix

1.7 Some Special Theorems

1.8 Complex Matrices

Exercises

References

Chapter II Groups and Jacobian of Some Transformations

2.0 Introduction

2.1 Groups

2.2 Some Examples of Groups

2.3 Normal Subgroups, Quotient Group, Homomorphism, Isomorphism, Direct Product

2.4 Jacobian of Some Transformations

Further Reading

Chapter III Notions of Multivariate Distributions and Invariance in Statistical Inference

3.0 Introduction

3.1 Multivariate Distributions

3.2 Invariance in Statistical Testing of Hypotheses

3.3 Sufficiency and Invariance

3.4 Unbiasedness and Invariance

3.5 Invariance and Optimum Tests

3.6 Most Stringent Tests and Invariance

Exercises

References

Chapter IV Multivariate Normal Distribution, Its Properties and Characterization

4.0 Introduction

4.1 Multivariate Normal Distribution (Classical Approach)

4.2 Some Characterizations of the Normal Distribution

4.3 Complex Multivariate Normal Distribution

4.4 Concentration Ellipsoid and Axes

4.5 Some Examples

4.6 Regression, Multiple and Partial Correlation

Exercises

References

Chapter V Estimators of Parameters and Their Functions in a Multivariate Normal Distribution

5.0 Introduction

5.1 Maximum Likelihood Estimators of μ, Σ

5.2 Properties of Maximum Likelihood Estimators of μ and Σ

5.3 Bayes, Minimax, and Admissible Characters of the Maximum Likelihood Estimator of μ, Σ

Exercises

References

Chapter VI Basic Multivariate Sampling Distributions

6.0 Introduction

6.1 Noncentral Chi-Square, Student's t-, F-Distributions

6.2 Distribution of Quadratic Forms, Cochran's Theorem

6.3 The Wishart Distribution

6.4 Tensor Product, Properties of the Wishart Distribution

6.5 The Noncentral Wishart Distribution

6.6 Generalized Variance

6.7 Distribution of the Bartlett Decomposition (Rectangular Coordinates)

6.8 Distribution of Hotelling's T2 and a Related Distribution

6.9 Distribution of Multiple and Partial Correlation Coefficients

Exercises

References

Chapter VII Tests of Hypotheses of Mean Vectors

7.0 Introduction

7.1 Tests and Confidence Region for Mean Vectors with Known Covariance Matrices

7.2 Tests and Confidence Region for Mean Vectors with Unknown Covariance Matrices

7.3 Tests of Hypotheses Concerning Subvectors of μ

Exercises

References

Chapter VIII Tests Concerning Covariance Matrices and Mean Vectors

8.0 Introduction

8.1 Hypothesis that a Covariance Matrix is Equal to a Given Matrix

8.2 The Sphericity Test

8.3 Tests of Independence and the R2-Test

8.4 Multivariate General Linear Hypothesis

8.5 Equality of Several Covariance Matrices

Exercises

References

Chapter IX Discriminant Analysis

9.0 Introduction

9.1 Examples

9.2 Formulation of the Problem of Discriminant Analysis

9.3 Classification into One of Two Multivariate Normal Populations

9.4 Classification into More than Two Multivariate Normal Populations

9.5 Concluding Remarks

Exercises

References

Chapter X Multivariate Covariance Models

10.0 Introduction

10.1 Principal Components

10.2 Factor Analysis

10.3 Canonical Correlations

10.4 Time Series Analysis

Exercises

References

Indexes

## Details

- No. of pages:
- 336

- Language:
- English

- Copyright:
- © Academic Press 1977

- Published:
- 1st January 1977

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483263335

## About the Author

### Narayan C. Giri

## About the Editor

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University