Theory of Statistics

Content.- 1: Probability Models.- 1.1 Background.- 1.2 Exchangeability.- 1.4 DeFinetti’s Representation Theorem.- 1.5 Proofs of DeFinetti’s Theorem and Related Results*.- 1.6 Infinite-Dimensional Parameters*.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.2 Exponential Families of Distributions.- 2.4 Extremal Families*.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.2 Classical Decision Theory.- 3.3 Axiomatic Derivation of Decision Theory*.- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.2 Bayesian Solutions.- 4.3 Most Powerful Tests.- 4.4 Unbiased Tests.- 4.5 Nuisance Parameters.- 4.6 P-Values.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.2 Set Estimation.- 5.3 The Bootstrap*.- 5.4 Problems.- 6: Equivariance*.- 6.1 Common Examples.- 6.2 Equivariant Decision Theory.- 6.3 Testing and Confidence Intervals*.- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.2 Sample Quantiles.- 7.3 Large Sample Estimation.- 7.4 Large Sample Properties of Posterior Distributions.- 7.5 Large Sample Tests.- 7.6 Problems.- 8: Hierarchical Models.- 8.1 Introduction.- 8.3 Nonnormal Models*.- 8.4 Empirical Bayes Analysis*.- 8.5 Successive Substitution Sampling.- 8.6 Mixtures of Models.- 8.7 Problems.- 9: Sequential Analysis.- 9.1 Sequential Decision Problems.- 9.2 The Sequential Probability Ratio Test.- 9.3 Interval Estimation*.- 9.4 The Relevancc of Stopping Rules.- 9.5 Problems.- Appendix A: Measure and Integration Theory.- A.1 Overview.- A.1.1 Definitions.- A.1.2 Measurable Functions.- A.1.3 Integration.- A.1.4 Absolute Continuity.- A.2 Measures.- A.3 Measurable Functions.- A.4 Integration.- A.5 Product Spaces.- A.6 Absolute Continuity.- A.7 Problems.- Appendix B: Probability Theory.- B.1 Overview.- B.1.1Mathematical Probability.- B.1.2 Conditioning.- B.1.3 Limit Theorems.- B.2 Mathematical Probability.- B.2.1 Random Quantities and Distributions.- B.2.2 Some Useful Inequalities.- B.3 Conditioning.- B.3.1 Conditional Expectations.- B.3.2 Borel Spaces*.- B.3.3 Conditional Densities.- B.3.4 Conditional Independence.- B.3.5 The Law of Total Probability.- B.4 Limit Theorems.- B.4.1 Convergence in Distribution and in Probability.- B.4.2 Characteristic Functions.- B.5 Stochastic Processes.- B.5.1 Introduction.- B.5.3 Markov Chains*.- B.5.4 General Stochastic Processes.- B.6 Subjective Probability.- B.7 Simulation*.- B.8 Problems.- Appendix C: Mathematical Theorems Not Proven Here.- C.1 Real Analysis.- C.2 Complex Analysis.- C.3 Functional Analysis.- Appendix D: Summary of Distributions.- D.1 Univariate Continuous Distributions.- D.2 Univariate Discrete Distributions.- D.3 Multivariate Distributions.- References.- Notation and Abbreviation Index.- Name Index.