Elementary Decision Theory

Title Page; Copyright Page; Dedication; Preface; Acknowledgments; Table of Contents; CHAPTER 1 -- Introduction; 1. INTRODUCTION; 2. AN EXAMPLE; 3. PRINCIPLES USED IN DECISION MAKING; 4. SUMMARY; SUGGESTED READINGS; CHAPTER 2 -- Data Processing; 1. INTRODUCTION; 2. DATA REPRESENTATION; 3. GRAPHICAL REPRESENTATIONS: HISTOGRAM; 4. GRAPHICAL REPRESENTATIONS: CUMULATIVE FREQUENCY POLYGON; 5. DESCRIPTIVE MEASURES: SUMMATION; 6. DESCRIPTIVE MEASURES: SAMPLE MEAN AND VARIANCE; 7. SIMPLIFIED COMPUTATION SCHEME FOR SAMPLE MEAN AND STANDARD DEVIATION USING GROUPED DATA; 8. SUMMARY; SUGGESTED READINGS.;This well-known and highly respected introduction to decision theory was developed at Stanford University. It furnishes a simple and clear-cut method of exhibiting the fundamental aspects of statistical problems. Beginners will find this treatment a motivating introduction to important mathematical notions such as set, function, and convexity. Title Page......Page 2 Copyright Page......Page 3 Dedication......Page 5 Preface......Page 6 Acknowledgments......Page 9 Table of Contents......Page 11 1. INTRODUCTION......Page 12 2. AN EXAMPLE......Page 14 3. PRINCIPLES USED IN DECISION MAKING......Page 23 4. SUMMARY......Page 24 SUGGESTED READINGS......Page 33 2. DATA REPRESENTATION......Page 34 3. GRAPHICAL REPRESENTATIONS: HISTOGRAM......Page 38 4. GRAPHICAL REPRESENTATIONS: CUMULATIVE FREQUENCY POLYGON......Page 41 5. DESCRIPTIVE MEASURES: SUMMATION......Page 47 6. DESCRIPTIVE MEASURES: SAMPLE MEAN AND VARIANCE......Page 54 7. SIMPLIFIED COMPUTATION SCHEME FOR SAMPLE MEAN AND STANDARD DEVIATION USING GROUPED DATA......Page 61 8. SUMMARY......Page 63 SUGGESTED READINGS......Page 64 2. TWO EXAMPLES......Page 65 3. PROBABILITY DISTRIBUTIONS AND CUMULATIVE DISTRIBUTION FUNCTIONS......Page 71 4. PROBABILITY DENSITY FUNCTION—DISCRETE CASE......Page 80 5. PROBABILITY DENSITY FUNCTION-CONTINUOUS CASE......Page 83 †6. POPULATION AND RANDOM SAMPLES......Page 86 7. THE NORMAL POPULATION......Page 91 8. SETS AND FUNCTION......Page 98 9. REVIEW OF PROBABILITY......Page 104 SUGGESTED READING......Page 112 2. UTILITY......Page 113 3. PROBABILITY AND EXPECTATION......Page 127 †4. APPLICATION OF UTILITY TO FAIR BETS......Page 142 †5. THE ST. PETERSBURG PARADOX......Page 148 6. DESCRIPTIVE PARAMETERS......Page 151 7. THE MEAN AND VARIANCE......Page 161 8. SUMMARY......Page 166 SUGGESTED READINGS......Page 168 2. TWO STATES OF NATURE—AN EXAMPLE......Page 170 3. TWO STATES OF NATURE: CONVEX SETS AND LINES......Page 180 4. TWO STATES OF NATURE: BAYES STRATEGIES AND SUPPORTING LINES......Page 191 5. TWO STATES OF NATURE: MINIMAX STRATEGIES......Page 207 6. TWO STATES OF NATURE: REGRET......Page 210 7. LINES, PLANES, AND CONVEX SETS IN HIGHER DIMENSIONS......Page 215 8. THREE OR MORE UNKNOWN STATES OF NATURE......Page 222 9. SUMMARY......Page 227 SUGGESTED READINGS......Page 231 1. A POSTERIORI PROBABILITY AND THE NO-DATA PROBLEM......Page 233 2. CONDITIONAL PROBABILITY......Page 238 3. A POSTERIORI PROBABILITY......Page 244 4. COMPUTATION OF BAYES STRATEGIES......Page 250 5. INDEPENDENCE......Page 257 6. SUMMARY......Page 266 7. REVIEW AT THE END OF CHAPTER 6......Page 270 SUGGESTED READINGS......Page 273 2. AN EXAMPLE OF HYPOTHESIS TESTING......Page 275 3. ESTIMATION......Page 288 4. CONFIDENCE INTERVALS......Page 296 †5. SIGNIFICANCE TESTING......Page 301 †6. A DECISION MAKING PROBLEM WHERE COMMON STATISTICAL PROCEDURES DO NOT APPLY......Page 307 7. SUMMARY......Page 311 1. INTRODUCTION......Page 316 2. MODELS OF PROBABILITY AND UTILITY......Page 318 3. MODELS OF THE SET OF AVAILABLE ACTIONS......Page 320 4. MODELS OF SETS OF POSSIBLE STATES OF NATURE......Page 323 5. MODELS OF REGRET FUNCTIONS......Page 324 6. MODELS OF EXPERIMENTS......Page 327 7. MODELS OF THE SET OF AVAILABLE STRATEGIES......Page 334 8. THE MODELS FOR THE PROBLEMS OF TESTING AND ESTIMATION......Page 336 9. SUMMARY......Page 338 2. NOTATION......Page 340 3. SIMPLE HYPOTHESIS VERSUS SIMPLE HYPOTHESIS (TWO STATES OF NATURE)......Page 342 4. COMPOSITE HYPOTHESES INVOLVING ONE PARAMETER......Page 352 5. COMPOSITE HYPOTHESES INVOLVING ONE PARAMETER: TWO-TAILED TESTS......Page 357 6. SEVERAL PARAMETERS......Page 362 7. DESIGN OF EXPERIMENTS......Page 365 8. SEQUENTIAL ANALYSIS......Page 367 9. SUMMARY......Page 375 SUGGESTED READINGS......Page 378 1. INTRODUCTION......Page 379 2. FORMAL STRUCTURE OF THE ESTIMATION PROBLEM: ONE-PARAMETER CASE......Page 380 3. METHODS OF ESTIMATION......Page 382 4. LARGE-SAMPLE PROPERTIES OF ESTIMATORS......Page 390 5. SMALL-SAMPLE PROPERTIES OF ESTIMATORS......Page 397 6. SEVERAL PARAMETERS......Page 403 7. CONFIDENCE INTERVALS: LARGE SAMPLES......Page 404 8. CONFIDENCE INTERVALS: SMALL SAMPLES......Page 406 9. SUMMARY......Page 409 SUGGESTED READINGS......Page 412 APPENDIX A - Notation......Page 413 APPENDIX B1......Page 421 APPENDIX C1......Page 423 APPENDIX D1......Page 425 APPENDIX E1......Page 429 APPENDIX F1 - Remarks About Game Theory......Page 467 Partial List of Answers to Exercises......Page 476 Index......Page 483 Title Page 2 Copyright Page 3 Dedication 5 Preface 6 Acknowledgments 9 Table of Contents 11 CHAPTER 1 - Introduction 12 1. INTRODUCTION 12 2. AN EXAMPLE 14 3. PRINCIPLES USED IN DECISION MAKING 23 4. SUMMARY 24 SUGGESTED READINGS 33 CHAPTER 2 - Data Processing 34 1. INTRODUCTION 34 2. DATA REPRESENTATION 34 3. GRAPHICAL REPRESENTATIONS: HISTOGRAM 38 4. GRAPHICAL REPRESENTATIONS: CUMULATIVE FREQUENCY POLYGON 41 5. DESCRIPTIVE MEASURES: SUMMATION 47 6. DESCRIPTIVE MEASURES: SAMPLE MEAN AND VARIANCE 54 7. SIMPLIFIED COMPUTATION SCHEME FOR SAMPLE MEAN AND STANDARD DEVIATION USING GROUPED DATA 61 8. SUMMARY 63 SUGGESTED READINGS 64 CHAPTER 3 - Introduction to Probability and Random Variables 65 1. INTRODUCTION 65 2. TWO EXAMPLES 65 3. PROBABILITY DISTRIBUTIONS AND CUMULATIVE DISTRIBUTION FUNCTIONS 71 4. PROBABILITY DENSITY FUNCTION—DISCRETE CASE 80 5. PROBABILITY DENSITY FUNCTION-CONTINUOUS CASE 83 †6. POPULATION AND RANDOM SAMPLES 86 7. THE NORMAL POPULATION 91 8. SETS AND FUNCTION 98 9. REVIEW OF PROBABILITY 104 SUGGESTED READING 112 CHAPTER 4 - Utility and Descriptive Statistics 113 1. INTRODUCTION 113 2. UTILITY 113 3. PROBABILITY AND EXPECTATION 127 †4. APPLICATION OF UTILITY TO FAIR BETS 142 †5. THE ST. PETERSBURG PARADOX 148 6. DESCRIPTIVE PARAMETERS 151 7. THE MEAN AND VARIANCE 161 8. SUMMARY 166 SUGGESTED READINGS 168 CHAPTER 5 - Uncertainty due to Ignorance of the State of Nature 170 1. INTRODUCTION 170 2. TWO STATES OF NATURE—AN EXAMPLE 170 3. TWO STATES OF NATURE: CONVEX SETS AND LINES 180 4. TWO STATES OF NATURE: BAYES STRATEGIES AND SUPPORTING LINES 191 5. TWO STATES OF NATURE: MINIMAX STRATEGIES 207 6. TWO STATES OF NATURE: REGRET 210 7. LINES, PLANES, AND CONVEX SETS IN HIGHER DIMENSIONS 215 8. THREE OR MORE UNKNOWN STATES OF NATURE 222 9. SUMMARY 227 SUGGESTED READINGS 231 CHAPTER 6 - The Computation of Bayes Strategies 233 1. A POSTERIORI PROBABILITY AND THE NO-DATA PROBLEM 233 2. CONDITIONAL PROBABILITY 238 3. A POSTERIORI PROBABILITY 244 4. COMPUTATION OF BAYES STRATEGIES 250 5. INDEPENDENCE 257 6. SUMMARY 266 7. REVIEW AT THE END OF CHAPTER 6 270 SUGGESTED READINGS 273 CHAPTER 7 - Introduction to Classical Statistics 275 1. INTRODUCTION 275 2. AN EXAMPLE OF HYPOTHESIS TESTING 275 3. ESTIMATION 288 4. CONFIDENCE INTERVALS 296 †5. SIGNIFICANCE TESTING 301 †6. A DECISION MAKING PROBLEM WHERE COMMON STATISTICAL PROCEDURES DO NOT APPLY 307 7. SUMMARY 311 CHAPTER 8 - Models 316 1. INTRODUCTION 316 2. MODELS OF PROBABILITY AND UTILITY 318 3. MODELS OF THE SET OF AVAILABLE ACTIONS 320 4. MODELS OF SETS OF POSSIBLE STATES OF NATURE 323 5. MODELS OF REGRET FUNCTIONS 324 6. MODELS OF EXPERIMENTS 327 7. MODELS OF THE SET OF AVAILABLE STRATEGIES 334 8. THE MODELS FOR THE PROBLEMS OF TESTING AND ESTIMATION 336 9. SUMMARY 338 CHAPTER 9 - Testing Hypotheses 340 1. INTRODUCTION 340 2. NOTATION 340 3. SIMPLE HYPOTHESIS VERSUS SIMPLE HYPOTHESIS (TWO STATES OF NATURE) 342 4. COMPOSITE HYPOTHESES INVOLVING ONE PARAMETER 352 5. COMPOSITE HYPOTHESES INVOLVING ONE PARAMETER: TWO-TAILED TESTS 357 6. SEVERAL PARAMETERS 362 7. DESIGN OF EXPERIMENTS 365 8. SEQUENTIAL ANALYSIS 367 9. SUMMARY 375 SUGGESTED READINGS 378 CHAPTER 10 - Estimation and Confidence Intervals 379 1. INTRODUCTION 379 2. FORMAL STRUCTURE OF THE ESTIMATION PROBLEM: ONE-PARAMETER CASE 380 3. METHODS OF ESTIMATION 382 4. LARGE-SAMPLE PROPERTIES OF ESTIMATORS 390 5. SMALL-SAMPLE PROPERTIES OF ESTIMATORS 397 6. SEVERAL PARAMETERS 403 7. CONFIDENCE INTERVALS: LARGE SAMPLES 404 8. CONFIDENCE INTERVALS: SMALL SAMPLES 406 9. SUMMARY 409 SUGGESTED READINGS 412 APPENDIX A - Notation 413 APPENDIX B1 421 APPENDIX C1 423 APPENDIX D1 425 APPENDIX E1 429 APPENDIX F1 - Remarks About Game Theory 467 Partial List of Answers to Exercises 476 Index 483 EBC,Converted "The text is very clearly written [with] many illustrative examples and exercises [and] should be considered by those instructors who would like to introduce a more modern (and a more logical) approach in a basic course in statistics." — Journal of the American Statistical Association This volume is a well-known, well-respected introduction to a lively area of statistics. Professors Chernoff and Moses bring years of professional expertise as classroom teachers to this straightforward approach to statistical problems. And happily, for beginning students, they have by-passed involved computational reasonings which would only confuse the mathematical novice. Developed from nine years of teaching statistics at Stanford, the book furnishes a simple and clear-cut method of exhibiting the fundamental aspects of a statistical problem. Beginners will find this book a motivating introduction to important mathematical notions such as set, function and convexity. Examples and exercises throughout introduce new topics and ideas. The first seven chapters are recommended for beginning courses in the basic ideas of statistics and require only a knowledge of high school math. These sections include material on data processing, probability and random variables, utility and descriptive statistics, uncertainty due to ignorance of the state of nature, computing Bayes strategies and an introduction to classical statistics. The last three chapters review mathematical models and summarize terminology and methods of testing hypotheses. Tables and appendixes provide information on notation, shortcut computational formulas, axioms of probability, properties of expectations, likelihood ratio test, game theory, and utility functions. Authoritative, yet elementary in its approach to statistics and statistical theory, this work is also concise, well-indexed and abundantly equipped with exercise material. Ideal for a beginning course, this modestly priced edition will be especially valuable to those interested in the principles of statistics and scientific method. This well-respected introduction to statistics and statistical theory covers data processing, probability and random variables, utility and descriptive statistics, computation of Bayes strategies, models, testing hypotheses, and much more. 1959 edition.