Optimal Statistical Decisions

Optimal Statistical Decisions

by Morris H. DeGroot



Product Details

ISBN-13: 9780070162426
Publisher: McGraw-Hill Companies, The
Publication date: 01/28/1970
Pages: 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

Part 1Survey of probability theory
Chapter 1.Introduction3
Chapter 2.Experiments, Sample Spaces, and Probability6
2.1Experiments and Sample Spaces6
2.2Set Theory7
2.3Events and Probability9
2.4Conditional Probability11
2.5Binomial Coefficients12
Chapter 3.Random Variables, Random Vectors, and Distribution Functions16
3.1Random Variables and Their Distributions16
3.2Multivariate Distributions17
3.3Sums and Integrals19
3.4Marginal Distributions and Independence20
3.5Vectors and Matrices21
3.6Expectations, Moments, and Characteristic Functions23
3.7Transformations of Random Variables26
3.8Conditional Distributions28
Chapter 4.Some Special Univariate Distributions33
4.2The Bernoulli Distribution34
4.3The Binomial Distribution34
4.4The Poisson Distribution35
4.5The Negative Binomial Distribution35
4.6The Hypergeometric Distribution36
4.7The Normal Distribution37
4.8The Gamma Distribution39
4.9The Beta Distribution40
4.10The Uniform Distribution40
4.11The Pareto Distribution41
4.12The t Distribution41
4.13The F Distribution42
Chapter 5.Some Special Multivariate Distributions48
5.2The Multinomial Distribution48
5.3The Dirichlet Distribution49
5.4The Multivariate Normal Distribution51
5.5The Wishart Distribution56
5.6The Multivariate t Distribution59
5.7The Bilateral Bivariate Pareto Distribution62
Part 2Subjective probability and utility
Chapter 6.Subjective Probability69
6.2Relative Likelihood70
6.3The Auxiliary Experiment75
6.4Construction of the Probability Distribution77
6.5Verification of the Properties of a Probability Distribution78
6.6Conditional Likelihoods81
Chapter 7.Utility86
7.1Preferences among Rewards86
7.2Preferences among Probability Distributions88
7.3The Definition of a Utility Function90
7.4Some Properties of Utility Functions92
7.5The Utility of Monetary Rewards95
7.6Convex and Concave Utility Functions97
7.7The Axiomatic Development of Utility101
7.8Construction of the Utility Function103
7.9Verification of the Properties of a Utility Function106
7.10Extension of the Properties of a Utility Function to the Class P[subscript E]110
Part 3Statistical decision problems
Chapter 8.Decision Problems121
8.1Elements of a Decision Problem121
8.2Bayes Risk and Bayes Decisions123
8.3Nonnegative Loss Functions124
8.4Concavity of the Bayes Risk125
8.5Randomization and Mixed Decisions128
8.6Convex Sets130
8.7Decision Problems in Which [similar]2 and D Are Finite132
8.8Decision Problems with Observations136
8.9Construction of Bayes Decision Functions138
8.10The Cost of Observation142
8.11Statistical Decision Problems in Which Both [Omega] and D Contain Two Points146
8.12Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage147
Chapter 9.Conjugate Prior Distributions155
9.1Sufficient Statistics155
9.2Conjugate Families of Distributions159
9.3Construction of the Conjugate Family161
9.4Conjugate Families for Samples from Various Standard Distributions164
9.5Conjugate Families for Samples from a Normal Distribution166
9.6Sampling from a Normal Distribution with Unknown Mean and Unknown Precision168
9.7Sampling from a Uniform Distribution172
9.8A Conjugate Family for Multinomial Observations174
9.9Conjugate Families for Samples from a Multivariate Normal Distribution175
9.10Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix177
9.11The Marginal Distribution of the Mean Vector179
9.12The Distribution of a Correlation180
9.13Precision Matrices Having an Unknown Factor182
Chapter 10.Limiting Posterior Distributions190
10.1Improper Prior Distributions190
10.2Improper Prior Distributions for Samples from a Normal Distribution194
10.3Improper Prior Distributions for Samples from a Multivariate Normal Distribution196
10.4Precise Measurement198
10.5Convergence of Posterior Distributions201
10.7Solutions of the Likelihood Equation208
10.8Convergence of Supercontinuous Functions210
10.9Limiting Properties of the Likelihood Function212
10.10Normal Approximation to the Posterior Distribution215
10.11Approximations for Vector Parameters216
10.12Posterior Ratios220
Chapter 11.Estimation, Testing Hypotheses, and Linear Statistical Models226
11.2Quadratic Loss227
11.3Loss Proportional to the Absolute Value of the Error231
11.4Estimation of a Vector233
11.5Problems of Testing Hypotheses237
11.6Testing a Simple Hypothesis about the Mean of a Normal Distribution239
11.7Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown241
11.8Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value244
11.9Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value247
11.10Linear Models249
11.11Testing Hypotheses in Linear Models253
11.12Investigating the Hypothesis That Certain Regression Coefficients Vanish256
11.13One-way Analysis of Variance257
Part 4Sequential decisions
Chapter 12.Sequential Sampling267
12.1Gains from Sequential Sampling267
12.2Sequential Decision Procedures272
12.3The Risk of a Sequential Decision Procedure275
12.4Backward Induction277
12.5Optimal Bounded Sequential Decision Procedures278
12.6Illustrative Examples280
12.7Unbounded Sequential Decision Procedures287
12.8Regular Sequential Decision Procedures289
12.9Existence of an Optimal Procedure290
12.10Approximating an Optimal Procedure by Bounded Procedures294
12.11Regions for Continuing or Terminating Sampling297
12.12The Functional Equation300
12.13Approximations and Bounds for the Bayes Risk302
12.14The Sequential Probability-ratio Test306
12.15Characteristics of Sequential Probability-ratio Tests309
12.16Approximating the Expected Number of Observations313
Chapter 13.Optimal Stopping324
13.2The Statistician's Reward325
13.3Choice of the Utility Function327
13.4Sampling without Recall331
13.5Further Problems of Sampling with Recall and Sampling without Recall333
13.6Sampling without Recall from a Normal Distribution with Unknown Mean336
13.7Sampling with Recall from a Normal Distribution with Unknown Mean341
13.8Existence of Optimal Stopping Rules345
13.9Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall349
13.11Stopping Rules for Martingales356
13.12Uniformly Integrable Sequences of Random Variables359
13.13Martingales Formed from Sums and Products of Random Variables361
13.14Regular Supermartingales365
13.15Supermartingales and General Problems of Optimal Stopping368
13.16Markov Processes369
13.17Stationary Stopping Rules for Markov Processes372
13.18Entrance-fee Problems376
13.19The Functional Equation for a Markov Process377
Chapter 14.Sequential Choice of Experiments385
14.2Markovian Decision Processes with a Finite Number of Stages386
14.3Markovian Decision Processes with an Infinite Number of Stages388
14.4Some Betting Problems391
14.5Two-armed-bandit Problems394
14.6Two-armed-bandit Problems When the Value of One Parameter Is Known396
14.7Two-armed-bandit Problems When the Parameters Are Dependent399
14.8Inventory Problems405
14.9Inventory Problems with an Infinite Number of Stages408
14.10Control Problems411
14.11Optimal Control When the Process Cannot Be Observed without Error414
14.12Multidimensional Control Problems418
14.13Control Problems with Actuation Errors421
14.14Search Problems423
14.15Search Problems with Equal Costs427
14.16Uncertainty Functions and Statistical Decision Problems429
14.17Sufficient Experiments433
14.18Examples of Sufficient Experiments437
Supplementary Bibliography466
Name Index475
Subject Index481

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