# Optimal Statistical Decisions

\$90.65

## Product Details

ISBN-13: 9780070162426 McGraw-Hill Companies, The 01/28/1970 489

## About the Author

Morris DeGroot (now deceased) was a great statistician and gentleman of the 20th Century. Says friend and coworker Joseph B. ("Jay") Kadane of DeGroot in the Foreword, "He was an institutional builder, as founder of the Statistics Department at Carnegie Mellon University and as first Executive Editor of Statistical Science. He was a wonderful colleague and friend, always ready for a chat about principles, a research problem, a departmental problem, a reference, or personal advice."

## Table of Contents

 Foreword vii Preface ix Part 1 Survey of probability theory Chapter 1. Introduction 3 Chapter 2. Experiments, Sample Spaces, and Probability 6 2.1 Experiments and Sample Spaces 6 2.2 Set Theory 7 2.3 Events and Probability 9 2.4 Conditional Probability 11 2.5 Binomial Coefficients 12 Exercises 13 Chapter 3. Random Variables, Random Vectors, and Distribution Functions 16 3.1 Random Variables and Their Distributions 16 3.2 Multivariate Distributions 17 3.3 Sums and Integrals 19 3.4 Marginal Distributions and Independence 20 3.5 Vectors and Matrices 21 3.6 Expectations, Moments, and Characteristic Functions 23 3.7 Transformations of Random Variables 26 3.8 Conditional Distributions 28 Exercises 30 Chapter 4. Some Special Univariate Distributions 33 4.1 Introduction 33 4.2 The Bernoulli Distribution 34 4.3 The Binomial Distribution 34 4.4 The Poisson Distribution 35 4.5 The Negative Binomial Distribution 35 4.6 The Hypergeometric Distribution 36 4.7 The Normal Distribution 37 4.8 The Gamma Distribution 39 4.9 The Beta Distribution 40 4.10 The Uniform Distribution 40 4.11 The Pareto Distribution 41 4.12 The t Distribution 41 4.13 The F Distribution 42 Exercises 43 Chapter 5. Some Special Multivariate Distributions 48 5.1 Introduction 48 5.2 The Multinomial Distribution 48 5.3 The Dirichlet Distribution 49 5.4 The Multivariate Normal Distribution 51 5.5 The Wishart Distribution 56 5.6 The Multivariate t Distribution 59 5.7 The Bilateral Bivariate Pareto Distribution 62 Exercises 63 Part 2 Subjective probability and utility Chapter 6. Subjective Probability 69 6.1 Introduction 69 6.2 Relative Likelihood 70 6.3 The Auxiliary Experiment 75 6.4 Construction of the Probability Distribution 77 6.5 Verification of the Properties of a Probability Distribution 78 6.6 Conditional Likelihoods 81 Exercises 82 Chapter 7. Utility 86 7.1 Preferences among Rewards 86 7.2 Preferences among Probability Distributions 88 7.3 The Definition of a Utility Function 90 7.4 Some Properties of Utility Functions 92 7.5 The Utility of Monetary Rewards 95 7.6 Convex and Concave Utility Functions 97 7.7 The Axiomatic Development of Utility 101 7.8 Construction of the Utility Function 103 7.9 Verification of the Properties of a Utility Function 106 7.10 Extension of the Properties of a Utility Function to the Class P[subscript E] 110 Exercises 115 Part 3 Statistical decision problems Chapter 8. Decision Problems 121 8.1 Elements of a Decision Problem 121 8.2 Bayes Risk and Bayes Decisions 123 8.3 Nonnegative Loss Functions 124 8.4 Concavity of the Bayes Risk 125 8.5 Randomization and Mixed Decisions 128 8.6 Convex Sets 130 8.7 Decision Problems in Which [similar]2 and D Are Finite 132 8.8 Decision Problems with Observations 136 8.9 Construction of Bayes Decision Functions 138 8.10 The Cost of Observation 142 8.11 Statistical Decision Problems in Which Both [Omega] and D Contain Two Points 146 8.12 Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage 147 Exercises 149 Chapter 9. Conjugate Prior Distributions 155 9.1 Sufficient Statistics 155 9.2 Conjugate Families of Distributions 159 9.3 Construction of the Conjugate Family 161 9.4 Conjugate Families for Samples from Various Standard Distributions 164 9.5 Conjugate Families for Samples from a Normal Distribution 166 9.6 Sampling from a Normal Distribution with Unknown Mean and Unknown Precision 168 9.7 Sampling from a Uniform Distribution 172 9.8 A Conjugate Family for Multinomial Observations 174 9.9 Conjugate Families for Samples from a Multivariate Normal Distribution 175 9.10 Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix 177 9.11 The Marginal Distribution of the Mean Vector 179 9.12 The Distribution of a Correlation 180 9.13 Precision Matrices Having an Unknown Factor 182 Exercises 183 Chapter 10. Limiting Posterior Distributions 190 10.1 Improper Prior Distributions 190 10.2 Improper Prior Distributions for Samples from a Normal Distribution 194 10.3 Improper Prior Distributions for Samples from a Multivariate Normal Distribution 196 10.4 Precise Measurement 198 10.5 Convergence of Posterior Distributions 201 10.6 Supercontinuity 204 10.7 Solutions of the Likelihood Equation 208 10.8 Convergence of Supercontinuous Functions 210 10.9 Limiting Properties of the Likelihood Function 212 10.10 Normal Approximation to the Posterior Distribution 215 10.11 Approximations for Vector Parameters 216 10.12 Posterior Ratios 220 Exercises 222 Chapter 11. Estimation, Testing Hypotheses, and Linear Statistical Models 226 11.1 Estimation 226 11.2 Quadratic Loss 227 11.3 Loss Proportional to the Absolute Value of the Error 231 11.4 Estimation of a Vector 233 11.5 Problems of Testing Hypotheses 237 11.6 Testing a Simple Hypothesis about the Mean of a Normal Distribution 239 11.7 Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown 241 11.8 Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value 244 11.9 Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value 247 11.10 Linear Models 249 11.11 Testing Hypotheses in Linear Models 253 11.12 Investigating the Hypothesis That Certain Regression Coefficients Vanish 256 11.13 One-way Analysis of Variance 257 Exercises 260 Part 4 Sequential decisions Chapter 12. Sequential Sampling 267 12.1 Gains from Sequential Sampling 267 12.2 Sequential Decision Procedures 272 12.3 The Risk of a Sequential Decision Procedure 275 12.4 Backward Induction 277 12.5 Optimal Bounded Sequential Decision Procedures 278 12.6 Illustrative Examples 280 12.7 Unbounded Sequential Decision Procedures 287 12.8 Regular Sequential Decision Procedures 289 12.9 Existence of an Optimal Procedure 290 12.10 Approximating an Optimal Procedure by Bounded Procedures 294 12.11 Regions for Continuing or Terminating Sampling 297 12.12 The Functional Equation 300 12.13 Approximations and Bounds for the Bayes Risk 302 12.14 The Sequential Probability-ratio Test 306 12.15 Characteristics of Sequential Probability-ratio Tests 309 12.16 Approximating the Expected Number of Observations 313 Exercises 317 Chapter 13. Optimal Stopping 324 13.1 Introduction 324 13.2 The Statistician's Reward 325 13.3 Choice of the Utility Function 327 13.4 Sampling without Recall 331 13.5 Further Problems of Sampling with Recall and Sampling without Recall 333 13.6 Sampling without Recall from a Normal Distribution with Unknown Mean 336 13.7 Sampling with Recall from a Normal Distribution with Unknown Mean 341 13.8 Existence of Optimal Stopping Rules 345 13.9 Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall 349 13.10 Martingales 353 13.11 Stopping Rules for Martingales 356 13.12 Uniformly Integrable Sequences of Random Variables 359 13.13 Martingales Formed from Sums and Products of Random Variables 361 13.14 Regular Supermartingales 365 13.15 Supermartingales and General Problems of Optimal Stopping 368 13.16 Markov Processes 369 13.17 Stationary Stopping Rules for Markov Processes 372 13.18 Entrance-fee Problems 376 13.19 The Functional Equation for a Markov Process 377 Exercises 379 Chapter 14. Sequential Choice of Experiments 385 14.1 Introduction 385 14.2 Markovian Decision Processes with a Finite Number of Stages 386 14.3 Markovian Decision Processes with an Infinite Number of Stages 388 14.4 Some Betting Problems 391 14.5 Two-armed-bandit Problems 394 14.6 Two-armed-bandit Problems When the Value of One Parameter Is Known 396 14.7 Two-armed-bandit Problems When the Parameters Are Dependent 399 14.8 Inventory Problems 405 14.9 Inventory Problems with an Infinite Number of Stages 408 14.10 Control Problems 411 14.11 Optimal Control When the Process Cannot Be Observed without Error 414 14.12 Multidimensional Control Problems 418 14.13 Control Problems with Actuation Errors 421 14.14 Search Problems 423 14.15 Search Problems with Equal Costs 427 14.16 Uncertainty Functions and Statistical Decision Problems 429 14.17 Sufficient Experiments 433 14.18 Examples of Sufficient Experiments 437 Exercises 439 References 447 Supplementary Bibliography 466 Name Index 475 Subject Index 481

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