# Mathematical Basis of Statistics

## 1st Edition

### Probability and Mathematical Statistics: A Series of Monographs and Textbooks

**Authors:**Jean-René Barra

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

**eBook ISBN:**9781483191447

**Imprint:**Academic Press

**Published Date:**28th September 1981

**Page Count:**268

## Description

Mathematical Basis of Statistics provides information pertinent to the methods and the mathematical basis of statistics. This book discusses the fundamental notion of statistical space.

Organized into 12 chapters, this book begins with an overview of the notion of statistical space in mathematical statistics and discusses other analogies with probability theory. This text then presents the notions of sufficiency and freedom, which are fundamental and useful in statistics but do not correspond to any notion in probability theory. Other chapters consider the theory of nonsequential tests and explain the practical meaning of the mathematical tools employed in statistics. This book discusses as well distributions used most frequently in classical statistical problems based on the normal distribution and provides relationships among these distributions. The final chapter deals with certain problems of mathematical statistics that are related to various problems of functional analysis.

This book is a valuable resource for graduate and postgraduate students.

## Table of Contents

Foreword

Editor's Preface

Preface

Notation and Terminology

Chapter 1 Statistical Spaces

1. Statistical Spaces; the Dominated Case

2. Statistics, Integrable Statistics, Completeness

3. Prior and Posterior Distributions

4. Products of Statistical Spaces

Exercises

Chapter 2 Sufficiency and Freedom

1. Sufficient σ-Fields and Sufficient Statistics

2. Factorization Criterion of Sufficiency

3. Projection of a Statistic

4. Free Subfields and Distribution-Free Statistics

5. P-Minimum Sufficient Subfields and Statistics

6. Relationship Among Freedom, Completeness, Sufficiency, and Stochastic Independence

7. Existence of Free Events

Exercises

Chapter 3 Statistical Information

1. Introduction

2. Information (According to Fisher)

Exercises

Chapter 4 Statistical Inference

1. Introduction

2. Decisions and Strategies

3. Hypothesis Testing and Statistical Estimation

4. Choosing a Strategy

5. Quasi-Ordering Induced by a Loss Function

6. Nuisance Parameters

Exercises

Chapter 5 Testing Statistical Hypotheses

1. Definitions and Preliminary Remarks

2. Quasi-Ordering on Tests of Hypotheses

3. Optimal Tests

4. The Fundamental Neyman-Pearson Lemma

5. Determining Optimal Tests

6. Nonoptimal Methods

Exercises

Chapter 6 Statistical Estimation

1. Unbiased Estimators

2. Optimal Estimators

3. Construction of Confidence Regions

4. Optimal Set Estimators

Exercises

Chapter 7 The Multivariate Normal Distribution

1. Some Useful Distributions

2. Multivariate Normal Distributions

3. Quadratic Forms of Normal Vectors

4. Stochastic Dependence Among Normal Vectors

Exercises

Chapter 8 Random Matrices

1. Notation

2. Covariance and Characteristic Function of a Random Matrix

3. Some Miscellaneous Results

4. Fundamental Results

5. Normal Random Matrix

6. Generalized Gamma Distributions

7. Bartlett's Decomposition of a Gamma Distribution

8. Nonsingular Gamma Distribution

9. Generalized Beta Distributions

10. Generalized Noncentral Gamma Distributions

Exercises

Chapter 9 Linear-Normal Statistical Spaces

1. The Cochran Theorem

2. Linear-Normal Statistical Spaces

3. Fundamental Theorems

4. Testing Linear Hypotheses

5. Estimation of Linear Functions

6. Fundamental Lemmas for Analysis of Variance

7. Methodology of Analysis of Variance (Model I) in Experimental Design

8. Analysis of Variance for Some Standard Experimental Designs

9. Introduction to Analysis of Variance (Model II)

10. Generalized Linear-Normal Statistical Spaces

Exercises

Chapter 10 Exponential Statistical Spaces

1. Laplace Transform of a Measure

2. Analytical Properties of Exponential Statistical Spaces

3. Sufficient Statistics on an Exponential Statistical Space

4. Incomplete Exponential Statistical Spaces

5. The Behrens-Fisher Problem

Exercises

Chapter 11 Testing Hypotheses on Exponential Statistical Spaces

1. The Case of One Unknown Scalar Parameter

2. One-Sided and Two-Sided Tests on a One-Dimensional Exponential Statistical Space

3. Optimal Tests on a One-Dimensional Exponential Statistical Space

4. Testing Hypotheses in the Case of Nuisance Parameters

Exercises

Chapter 12 Functional Analysis and Mathematical Statistics

1. Computation of a Statistic Having a Given Image

2. Sufficient Statistics Having Minimum Dimension

3. Spaces of Statistics

4. Existence of Free Events

5. The Converse of the Neyman-Pearson Lemma

6. A Theorem of Linnik

Exercises

Appendix Conditional Probability

1. Preliminary Results

2. Conditional Expectation (with Respect to a σ-Field or a Random Element)

3. Conditional Probability (with Respect to a σ-Field or a Random Element)

4. Properties of Conditional Expectation

5. Conditional Distribution

6. Transition Probabilities

References

Index

## Details

- No. of pages:
- 268

- Language:
- English

- Copyright:
- © Academic Press 1981

- Published:
- 28th September 1981

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483191447

## About the Author

### Jean-René Barra

## About the Editor

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University