Mathematical Statistics: A Decision Theoretic Approach

Title......Page 1
Preface......Page 2
Contents......Page 6
1.1 Basic Elements......Page 10
1.2 A Comparison of Game Theory and Decision Theory......Page 14
1.3 Decision Function; Risk Function......Page 15
1.4 Utility and Subjective Probability......Page 20
1.5 Randomization......Page 31
1.6 Optimal Decision Rules......Page 37
1.7 Geometric Interpretation for Finite......Page 43
1.8 The Form of Bayes Rules for Estimation Problems......Page 52
2.1 Admissibility and Completeness......Page 63
2.2 Decision Theory......Page 65
2.3 Admissibility of Bayes Rules......Page 68
2.4 Basic Assumptions......Page 72
2.5 Existence of Bayes Decision Rules......Page 76
2.6 Existence of a Minimal Complete Class......Page 78
2.7 The Separating Hyperplane Theorem......Page 79
2.8 Essential Completeness of the Class of Nonrandomized Decision Rules......Page 85
2.9 The Minimax Theorem......Page 90
2.10 The Complete Class Theorem......Page 95
2.11 Solving for Minimax Rules......Page 99
3.1 Useful Univariate Distributions......Page 107
3.2 The Multivariate Normal Distribution......Page 114
3.3 Sufficient Statistics......Page 121
3.4 Essentially Complete Classes of Rules Based on Sufficient Statistics......Page 128
3.5 Exponential Families of Distributions......Page 134
3.6 Complete Sufficient Statistics......Page 141
3.7 Continuity of the Risk Function......Page 146
4.1 Invariant Decision Problems......Page 152
4.2 Invariant Decision Rules......Page 157
4.3 Admissible and Minimax Invariant Rules......Page 163
4.4 Location and Scale Parameters......Page 173
4.5 Minimax Estimates of Location Parameters......Page 175
4.6 Minimax Estimates for the Parameters of a Normal Distribution......Page 185
4.7 The Pitman Estimate......Page 195
4.8 Estimation of a Distribution Function......Page 200
5.1 The Neyman-Pearson Lemma......Page 207
5.2 Uniformly Most Powerful Tests......Page 215
5.3 Two-Sided Tests......Page 224
5.4 Uniformly Most Powerful Unbiased Tests......Page 233
5.5 Locally Best Tests......Page 244
5.6 Invariance in Hypothesis Testing......Page 251
5.7 The Two-Sample Problem......Page 259
5.8 Confidence Sets......Page 266
5.9 The General Linear Hypothesis......Page 273
5.10 Confidence Ellipsoids and Multiple Comparisons......Page 283
6.1 Monotone Multiple Decision Problems......Page 293
6.2 Bayes Rules in Multiple Decision Problems......Page 300
6.3 Slippage Problems......Page 308
7.1 Sequential Decision Rules......Page 318
7.2 Bayes and Minimax Sequential Decision Rules......Page 322
7.3 Convex Loss and Sufficiency......Page 338
7.4 Invariant Sequential Decision Problems......Page 349
7.5 Sequential Tests of a Simple Hypothesis Against a Simple Alternative......Page 359
7.6 The Sequential Probability Ratio Test......Page 370
7.7 The Fundamental Identity of Sequential Analysis......Page 379
References......Page 397
Subject Index......Page 402