Mathematical Statistics

PrefaceChapter 1. Game Theory and Decision Theory 1.1 Basic Elements 1.2 A Comparison of Game Theory and Decision Theory 1.3 Decision Function; Risk Function 1.4 Utility and Subjective Probability 1.5 Randomization 1.6 Optimal Decision Rules 1.7 Geometric Interpretation for Finite θ 1.8 The Form of Bayes Rules for Estimation ProblemsChapter 2. The Main Theorems of Decision Theory 2.1 Admissibility and Completeness 2.2 Decision Theory 2.3 Admissibility of Bayes Rules 2.4 Basic Assumptions 2.5 Existence of Bayes Decision Rules 2.6 Existence of a Minimal Complete Class 2.7 The Separating Hyperplane Theorem 2.8 Essential Completeness of the Class of Nonrandomized Decision Rules 2.9 The Minimax Theorem 2.10 The Complete Class Theorem 2.11 Solving for Minimax RulesChapter 3. Distributions and Sufficient Statistics 3.1 Useful Univariate Distributions 3.2 The Multivariate Normal Distribution 3.3 Sufficient Statistics Sufficient Statistics 3.5 Exponential Families of Distributions 3.6 Complete Sufficient Statistics 3.7 Continuity of the Risk FunctionChapter 4. Invariant Statistical Decision Problems 4.1 Invariant Decision Problems 4.2 Invariant Decision Rules 4.3 Admissible and Minimax Invariant Rules 4.4 Location and Scale Parameters 4.5 Minimax Estimates of Location Parameters 4.6 Minimax Estimates for the Parameters of a Normal Distribution 4.7 The Pitman Estimate 4.8 Estimation of a Distribution FunctionChapter 5. Testing Hypotheses 5.1 The Neyman-Pearson Lemma 5.2 Uniformly Most Powerful Tests 5.3 Two-Sided Tests 5.4 Uniformly Most Powerful Unbiased Tests 5.5 Locally Best Tests 5.6 Invariance in Hypothesis Testing 5.7 The Two-Sample Problem 5.8 Confidence Sets 5.9 The General Linear Hypothesis 5.10 Confidence Ellipsoids and Multiple ComparisonsChapter 6. Multiple Decision Problems 6.1 Monotone Multiple Decision Problems 6.2 Bayes Rules in Multiple Decision Problems 6.3 Slippage ProblemsChapter 7. Sequential Decision Problems 7.1 Sequential Decision Rules 7.2 Bayes and Minimax Sequential Decision Rules 7.3 Convex Loss and Sufficiency 7.4 Invariant Sequential Decision Problems 7.5 Sequential Tests of a Simple Hypothesis Against a Simple Alternative 7.6 The Sequential Probability Ratio Test 7.7 The Fundamental Identity of Sequential AnalysisReferencesSubject Index