Optimal Statistical Decisions

by DeGroot, Morris H.

• ISBN13: 9780471680291
• ISBN10: 047168029X
• Edition: 1st
• Format: Paperback
• Copyright: 2004-04-30
• Publisher: Wiley-Interscience

Summary

The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists.

Author Biography

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	p. vii
Preface	p. ix
Survey of probability theory
Introduction	p. 3
Experiments, Sample Spaces, and Probability	p. 6
Experiments and Sample Spaces	p. 6
Set Theory	p. 7
Events and Probability	p. 9
Conditional Probability	p. 11
Binomial Coefficients	p. 12
Exercises	p. 13
Random Variables, Random Vectors, and Distribution Functions	p. 16
Random Variables and Their Distributions	p. 16
Multivariate Distributions	p. 17
Sums and Integrals	p. 19
Marginal Distributions and Independence	p. 20
Vectors and Matrices	p. 21
Expectations, Moments, and Characteristic Functions	p. 23
Transformations of Random Variables	p. 26
Conditional Distributions	p. 28
Exercises	p. 30
Some Special Univariate Distributions	p. 33
Introduction	p. 33
The Bernoulli Distribution	p. 34
The Binomial Distribution	p. 34
The Poisson Distribution	p. 35
The Negative Binomial Distribution	p. 35
The Hypergeometric Distribution	p. 36
The Normal Distribution	p. 37
The Gamma Distribution	p. 39
The Beta Distribution	p. 40
The Uniform Distribution	p. 40
The Pareto Distribution	p. 41
The t Distribution	p. 41
The F Distribution	p. 42
Exercises	p. 43
Some Special Multivariate Distributions	p. 48
Introduction	p. 48
The Multinomial Distribution	p. 48
The Dirichlet Distribution	p. 49
The Multivariate Normal Distribution	p. 51
The Wishart Distribution	p. 56
The Multivariate t Distribution	p. 59
The Bilateral Bivariate Pareto Distribution	p. 62
Exercises	p. 63
Subjective probability and utility
Subjective Probability	p. 69
Introduction	p. 69
Relative Likelihood	p. 70
The Auxiliary Experiment	p. 75
Construction of the Probability Distribution	p. 77
Verification of the Properties of a Probability Distribution	p. 78
Conditional Likelihoods	p. 81
Exercises	p. 82
Utility	p. 86
Preferences among Rewards	p. 86
Preferences among Probability Distributions	p. 88
The Definition of a Utility Function	p. 90
Some Properties of Utility Functions	p. 92
The Utility of Monetary Rewards	p. 95
Convex and Concave Utility Functions	p. 97
The Axiomatic Development of Utility	p. 101
Construction of the Utility Function	p. 103
Verification of the Properties of a Utility Function	p. 106
Extension of the Properties of a Utility Function to the Class P[subscript E]	p. 110
Exercises	p. 115
Statistical decision problems
Decision Problems	p. 121
Elements of a Decision Problem	p. 121
Bayes Risk and Bayes Decisions	p. 123
Nonnegative Loss Functions	p. 124
Concavity of the Bayes Risk	p. 125
Randomization and Mixed Decisions	p. 128
Convex Sets	p. 130
Decision Problems in Which [similar]2 and D Are Finite	p. 132
Decision Problems with Observations	p. 136
Construction of Bayes Decision Functions	p. 138
The Cost of Observation	p. 142
Statistical Decision Problems in Which Both [Omega] and D Contain Two Points	p. 146
Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage	p. 147
Exercises	p. 149
Conjugate Prior Distributions	p. 155
Sufficient Statistics	p. 155
Conjugate Families of Distributions	p. 159
Construction of the Conjugate Family	p. 161
Conjugate Families for Samples from Various Standard Distributions	p. 164
Conjugate Families for Samples from a Normal Distribution	p. 166
Sampling from a Normal Distribution with Unknown Mean and Unknown Precision	p. 168
Sampling from a Uniform Distribution	p. 172
A Conjugate Family for Multinomial Observations	p. 174
Conjugate Families for Samples from a Multivariate Normal Distribution	p. 175
Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix	p. 177
The Marginal Distribution of the Mean Vector	p. 179
The Distribution of a Correlation	p. 180
Precision Matrices Having an Unknown Factor	p. 182
Exercises	p. 183
Limiting Posterior Distributions	p. 190
Improper Prior Distributions	p. 190
Improper Prior Distributions for Samples from a Normal Distribution	p. 194
Improper Prior Distributions for Samples from a Multivariate Normal Distribution	p. 196
Precise Measurement	p. 198
Convergence of Posterior Distributions	p. 201
Supercontinuity	p. 204
Solutions of the Likelihood Equation	p. 208
Convergence of Supercontinuous Functions	p. 210
Limiting Properties of the Likelihood Function	p. 212
Normal Approximation to the Posterior Distribution	p. 215
Approximations for Vector Parameters	p. 216
Posterior Ratios	p. 220
Exercises	p. 222
Estimation, Testing Hypotheses, and Linear Statistical Models	p. 226
Estimation	p. 226
Quadratic Loss	p. 227
Loss Proportional to the Absolute Value of the Error	p. 231
Estimation of a Vector	p. 233
Problems of Testing Hypotheses	p. 237
Testing a Simple Hypothesis about the Mean of a Normal Distribution	p. 239
Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown	p. 241
Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value	p. 244
Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value	p. 247
Linear Models	p. 249
Testing Hypotheses in Linear Models	p. 253
Investigating the Hypothesis That Certain Regression Coefficients Vanish	p. 256
One-way Analysis of Variance	p. 257
Exercises	p. 260
Sequential decisions
Sequential Sampling	p. 267
Gains from Sequential Sampling	p. 267
Sequential Decision Procedures	p. 272
The Risk of a Sequential Decision Procedure	p. 275
Backward Induction	p. 277
Optimal Bounded Sequential Decision Procedures	p. 278
Illustrative Examples	p. 280
Unbounded Sequential Decision Procedures	p. 287
Regular Sequential Decision Procedures	p. 289
Existence of an Optimal Procedure	p. 290
Approximating an Optimal Procedure by Bounded Procedures	p. 294
Regions for Continuing or Terminating Sampling	p. 297
The Functional Equation	p. 300
Approximations and Bounds for the Bayes Risk	p. 302
The Sequential Probability-ratio Test	p. 306
Characteristics of Sequential Probability-ratio Tests	p. 309
Approximating the Expected Number of Observations	p. 313
Exercises	p. 317
Optimal Stopping	p. 324
Introduction	p. 324
The Statistician's Reward	p. 325
Choice of the Utility Function	p. 327
Sampling without Recall	p. 331
Further Problems of Sampling with Recall and Sampling without Recall	p. 333
Sampling without Recall from a Normal Distribution with Unknown Mean	p. 336
Sampling with Recall from a Normal Distribution with Unknown Mean	p. 341
Existence of Optimal Stopping Rules	p. 345
Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall	p. 349
Martingales	p. 353
Stopping Rules for Martingales	p. 356
Uniformly Integrable Sequences of Random Variables	p. 359
Martingales Formed from Sums and Products of Random Variables	p. 361
Regular Supermartingales	p. 365
Supermartingales and General Problems of Optimal Stopping	p. 368
Markov Processes	p. 369
Stationary Stopping Rules for Markov Processes	p. 372
Entrance-fee Problems	p. 376
The Functional Equation for a Markov Process	p. 377
Exercises	p. 379
Sequential Choice of Experiments	p. 385
Introduction	p. 385
Markovian Decision Processes with a Finite Number of Stages	p. 386
Markovian Decision Processes with an Infinite Number of Stages	p. 388
Some Betting Problems	p. 391
Two-armed-bandit Problems	p. 394
Two-armed-bandit Problems When the Value of One Parameter Is Known	p. 396
Two-armed-bandit Problems When the Parameters Are Dependent	p. 399
Inventory Problems	p. 405
Inventory Problems with an Infinite Number of Stages	p. 408
Control Problems	p. 411
Optimal Control When the Process Cannot Be Observed without Error	p. 414
Multidimensional Control Problems	p. 418
Control Problems with Actuation Errors	p. 421
Search Problems	p. 423
Search Problems with Equal Costs	p. 427
Uncertainty Functions and Statistical Decision Problems	p. 429
Sufficient Experiments	p. 433
Examples of Sufficient Experiments	p. 437
Exercises	p. 439
References	p. 447
Supplementary Bibliography	p. 466
Name Index	p. 475
Subject Index	p. 481
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