Theory of Statistics

The aim of this graduate textbook is to provide a comprehensive advanced course in the theory of statistics covering those topics in estimation, testing, and large sample theory which a graduate student might typically need to learn as preparation for work on a Ph.D.
An important strength of this book is that it provides a mathematically rigorous account of both classical and Bayesian inference in order to give readers a broad perspective.
Commencing with chapters on probability models and the theory of sufficient statistics, the author covers decision theory, hypothesis testing, estimation, equivariance, large sample theory, hierarchical models, and, finally, sequential nalysis. Every chapter concludes with exercises which range in difficulty from the easy to the challenging. As a result, this textbook provides an excellent course in modern theoretical statistics.

Aims to provide a comprehensive advanced course in the theory of statistics covering those topics in estimation, testing, and large sample theory. This book provides an account of both Classical and Bayesian inference in order to give readers a broad perspective.

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.