Mathematical Statistics

Mathematical Statistics: A Decision Theoretic Approach presents an investigation of the extent to which problems of mathematical statistics may be treated by decision theory approach. This book deals with statistical theory that could be justified from a decision-theoretic viewpoint. Organized into seven chapters, this book begins with an overview of the elements of decision theory that are similar to those of the theory of games. This text then examines the main theorems of decision theory that involve two more notions, namely the admissibility of a decision rule and the completeness of a class of decision rules. Other chapters consider the development of theorems in decision theory that are valid in general situations. This book discusses as well the invariance principle that involves groups of transformations over the three spaces around which decision theory is built. The final chapter deals with sequential decision problems. This book is a valuable resource for first-year graduate students in mathematics.

Preface

Chapter 1. Game Theory and Decision Theory

1.1 Basic Elements

1.2 A Comparison of Game Theory and Decision Theory

1.3 Decision Function; Risk Function

1.4 Utility and Subjective Probability

1.5 Randomization

1.6 Optimal Decision Rules

1.7 Geometric Interpretation for Finite θ

1.8 The Form of Bayes Rules for Estimation Problems

Chapter 2. The Main Theorems of Decision Theory

2.1 Admissibility and Completeness

2.2 Decision Theory

2.3 Admissibility of Bayes Rules

2.4 Basic Assumptions

2.5 Existence of Bayes Decision Rules

2.6 Existence of a Minimal Complete Class

2.7 The Separating Hyperplane Theorem

2.8 Essential Completeness of the Class of Nonrandomized Decision Rules

2.9 The Minimax Theorem

2.10 The Complete Class Theorem

2.11 Solving for Minimax Rules

Chapter 3. Distributions and Sufficient Statistics

3.1 Useful Univariate Distributions

3.2 The Multivariate Normal Distribution

3.3 Sufficient Statistics Sufficient Statistics

3.5 Exponential Families of Distributions

3.6 Complete Sufficient Statistics

3.7 Continuity of the Risk Function

Chapter 4. Invariant Statistical Decision Problems

4.1 Invariant Decision Problems

4.2 Invariant Decision Rules

4.3 Admissible and Minimax Invariant Rules

4.4 Location and Scale Parameters

4.5 Minimax Estimates of Location Parameters

4.6 Minimax Estimates for the Parameters of a Normal Distribution

4.7 The Pitman Estimate

4.8 Estimation of a Distribution Function

Chapter 5. Testing Hypotheses

5.1 The Neyman-Pearson Lemma

5.2 Uniformly Most Powerful Tests

5.3 Two-Sided Tests

5.4 Uniformly Most Powerful Unbiased Tests

5.5 Locally Best Tests

5.6 Invariance in Hypothesis Testing

5.7 The Two-Sample Problem

5.8 Confidence Sets

5.9 The General Linear Hypothesis

5.10 Confidence Ellipsoids and Multiple Comparisons

Chapter 6. Multiple Decision Problems

6.1 Monotone Multiple Decision Problems

6.2 Bayes Rules in Multiple Decision Problems

6.3 Slippage Problems

Chapter 7. Sequential Decision Problems

7.1 Sequential Decision Rules

7.2 Bayes and Minimax Sequential Decision Rules

7.3 Convex Loss and Sufficiency

7.4 Invariant Sequential Decision Problems

7.5 Sequential Tests of a Simple Hypothesis Against a Simple Alternative

7.6 The Sequential Probability Ratio Test

7.7 The Fundamental Identity of Sequential Analysis

References

Subject Index