Elementary Decision Theory

Contents......Page 8
1 Introduction......Page 13
2 An Example......Page 14
3 Principles Used in Decision Making......Page 21
4 Summary......Page 22
2 Data Representation......Page 29
3 Graphical Representations: Histogram......Page 32
4 Graphical Representations: Cumulative Frequency Polygon......Page 35
5 Descriptive Measures: Summation......Page 41
6 Descriptive Measures: Sample Mean and Variance......Page 45
7 Simplified Computation Scheme for sample Mean and Standard Deviation Using Grouped Data......Page 50
8 Summary......Page 52
2 Two Examples......Page 53
3 Probability Distributions and Cumulative Distribution Functions......Page 58
4 Probability Density Function - Discrete Case......Page 66
5 Probability Density Function - Continuous Case......Page 68
6 Population and Random Samples......Page 71
7 The Normal Population......Page 74
8 Sets and Functions......Page 80
9 Review of Probability......Page 85
2 Utility......Page 91
2.1 The Assumptions Behind Utility......Page 93
2.2 Application of Utility......Page 98
3 Probability and Expectation......Page 101
4 Application of Utility to Fair Bets......Page 111
5 The St. Petersburg Paradox......Page 116
6 Descriptive Parameters......Page 118
7.1 Some Properties of the Mean, Variance, and Standard Deviation......Page 126
8 Summary......Page 129
2 Two States of Nature - an Example......Page 131
3 Two States of Nature: Convex Sets and Lines......Page 139
4 Two States of Nature: Bayes Strategies and Supporting Lines......Page 148
5 Two States of Nature: Minimax Strategies......Page 159
6 Two States of Nature: Regret......Page 162
7 Lines, Planes, And Convex Sets in Higher Dimensions......Page 167
8 Three or More Unknown States of Nature......Page 171
9 Summary......Page 175
1 A Posteriori Probability and the No-Data Problem......Page 178
2 Conditional Probability......Page 182
3 A Posteriori Probability......Page 186
4 Computation of Bayes Strategies......Page 191
4.1 Remarks......Page 192
5 Independence......Page 195
6 Summary......Page 202
7 Review at the End of Chapter 6......Page 204
2 An Example of Hypothesis Testing......Page 207
2.1 Bayes Strategies......Page 212
2.2 The Name \"Testing Hypotheses\"......Page 214
2.3 Another Example......Page 216
3 Estimation......Page 219
4 Confidence Intervals......Page 226
5 Significance Testing......Page 229
6 A Decision Making Problem Where Common Statistical Procedures Do Not Apply......Page 233
7 Summary......Page 237
1 Introduction......Page 240
2 Models of Probability and Utility......Page 242
3 Models of the Set of Available Actions......Page 243
4 Models of Sets of Possible States of Nature......Page 245
5 Models of Regret Functions......Page 246
6 Models of Experiments......Page 247
6.1 Specifications of Distributions......Page 248
6.2 Models Concerned with Relations Between Variables......Page 250
7 Models of the Set of Available Strategies......Page 252
8 The Models for the Problems of Testing and Estimation......Page 254
9 Summary......Page 255
2 Notation......Page 257
3 Simple Hypothesis Versus Simple Hypothesis (Two States of Nature)......Page 258
4 Composite Hypotheses Involving One Parameter......Page 267
5 Composite Hypotheses Involving One Parameter: Two-Tailed Tests......Page 270
6 Several Parameters......Page 274
7 Design of Experiments......Page 276
8 Sequential Analysis......Page 278
8.1 Details in the Binomial Example......Page 281
8.2 Details in the Normal Example with Known Variance......Page 283
9 Summary......Page 284
2 Formal Structure of the Estimation Problem: One-Parameter Case......Page 287
3.1 The Analogue Method or Method of Moments......Page 289
3.2 The Maximum-Likelihood Method......Page 290
3.3 Bayes Strategies......Page 292
3.4 Convenience of the Various Methods of Estimation......Page 294
4 Large-Sample Properties of Estimators......Page 295
5.1 Invariance Under Transformation of Parameter......Page 299
5.2 Unbiased Estimators......Page 300
5.3 Sufficiency......Page 302
6 Several Parameters......Page 303
7 Confidence Intervals: Large Samples......Page 304
8 Confidence Intervals: Small Samples......Page 305
9 Summary......Page 307
A Notation......Page 310
B1 Short Table of Squares and Square Roots......Page 314
B2 Logarithms......Page 315
C1 Table of Random Digits......Page 317
C2 Table of Random Normal Deviates......Page 318
D1 Areas Under the Normal Curve From \\mu + a \\sigma to \\infty......Page 319
D2 Chi Square Distribution......Page 320
D3 Exponential Distribution......Page 321
D4 Student\'s t Distribution......Page 322
E1 Derivation of an Equation for the Sample Variance......Page 323
E2 Simplified Computation Scheme for the Sample Mean and Variance\'Using Grouped Data......Page 324
E3 Axioms of Probability......Page 325
E4 Properties of Expectation......Page 327
E5 The Convex Set Generated by A is the Set of Weighted Averages of Elements of A......Page 333
E6 Relevance of Risks......Page 335
E7 Probabilities of Compound Sets......Page 337
E8 Bayes Strategies Obtained by Using A Posteriori Probabilities......Page 339
E9 Crossing Bridges One at a Time......Page 341
E10 Mean and Variance of X......Page 342
E11 For Testing a Simple Hypothesis Versus a Simple Alternative, the Bayes Strategies are Likelihood-Ratio Tests......Page 344
E12 Some Likelihood-Ratio Tests......Page 347
E13 Admissibility of the Two-State Likelihood-Ratio Tests for Certain Problems Involving Composite Hypotheses......Page 350
E14 Sequential Likelihood-Ratio Tests......Page 352
E15 Some Sequential Likelihood-Ratio Tests......Page 356
F1 Remarks About Game Theory......Page 358
F2 Outline of the Derivation of the Utility Function......Page 362
Partial List of Answers to Exercises......Page 365
Index......Page 370