The Foundations of Statistics

1. INTRODUCTION 1. The role of foundations 2. Historical background 3. General outline of this book 2. PRELIMINARY CONSIDERATIONS ON DECISION IN THE FACE OF UNCERTAINTY 1. Introduction 2. The person 3. "The world, and states of the world" 4. Events 5. "Consequences, acts, and decisions" 6. The simple ordering of acts with respect to preference 7. The sure-thing principle 3. PERSONAL PROBABILITY 1. Introduction 2. Qualitative personal probability 3. Quantitative personal probability 4. Some mathematical details 5. "Conditional probability, qualitative and quantitative" 6. The approach to certainty through experience 7. Symmetric sequences of events 4. CRITICAL COMMENTS ON PERSONAL PROBABILITY 1. Introduction 2. Some shortcomings of the personalistic view 3. Connection with other views 4. Criticism of other views 5. The role of symmetry in probability 6. How can science use a personalistic view of probability? 5. UTILITY 1. Introduction 2. Gambles 3. "Utility, and preference among gambles" 4. The extension of utility to more general acts 5. Small worlds 6. Historical and critical comments on utility 6. OBSERVATION 1. Introduction 2. What an observation is 3. "Multiple observations, and extensions of observations and of sets of acts" 4. Dominance and admissibility 5. Outline of the design of experiments 7. PARTITION PROBLEMS 1. Introduction 2. Structure of (twofold) partition problems 3. The value of observation 4. "Extension of observations, and sufficient statistics" 5. Likelihood ratios 6. Repeated observations 7. Sequential probability ratio procedures 8. "Standard form, and absolute comparison between observations" 8. STATISTICS PROPER 1. Introduction 2. What is statistics proper? 3. Multipersonal problems 4. The minimax theory 9. INTRODUCTION TO THE MINIMAX THEORY 1. Introduction 2. The behavioralistic outlook 3. Mixed acts 4. Income and loss 5. "The minimax rule, and the principle of admissibility" 6. Illustrations of the minimax rule 7. Objectivistic motivation of the minimax rule 8. Loss as opposed to negative income in the minimax rule 10. A PERSONALISTIC REINTERPRETATION OF THE MINIMAX THEORY 1. Introduction 2. A model of group decision 3. "The group minimax rule, and the group principle of admissibility" 4. Critique of the group minimax rule 11. THE PARALLELISM BETWEEN THE MINIMAX THEORY AND THE THEORY OF TWO-PERSON GAMES 1. Introduction 2. Standard games 3. Minimax play 4. Parallelism and contrast with the minimax theories 12. THE MATHEMATICS OF MINIMAX PROBLEMS 1. Introduction 2. Abstract games 3. Bilinear games 4. An example of a bilinear game 5. Bilinear games exhibiting symmetry 13. OJBECTIONS TO THE MINIMAX RULES 1. Introduction 2. A confusion between loss and negative income 3. Utility and the minimax rule 4. Almost sub-minimax acts 5. The minimax rule does not generate a simple ordering 14. THE MINIMAX THEORY APPLIED TO OBSERVATIONS 1. Introduction 2. Recapitulation of partition problems 3. Sufficient statistics 4. "Simple dichotomy, an example" 5. The approach to certainty 6. Cost of observation 7. Sequential probability ratio procedures 8. Randomization 9. Mixed acts in statistics 15. POINT ESTIMATION 1. Introduction 2. The verbalistic concept of point estimation 3. Examples of problems of point estimation 4. Criteria that have been proposed for point estimates 5. A behavioralistic review of the criteria for point estimation 6. "A behavioralistic review, continued" 7. "A behavioralistic review, concluded" 16. TESTING 1. Introduction 2. A theory of testing 3. Testing in practice 17. INTERVAL ESTIMATION AND RELATED TOPICS 1. Estimates of the accuracy of estimates 2. Interval estimation and confidence intervals 3. Tolerance intervals 4. Fiducial probability APPENDIX 1. EXPECTED VALUE APPENDIX 2. CONVEX FUNCTIONS APPENDIX 3. BIBLIOGRAPHIC MATERIAL APPENDIX 4. BIBLIOGRAPHIC SUPPLEMENT TECHNICAL SYMBOLS AUTHOR INDEX GENERAL INDEX