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 and even-handed account of both Classical and Bayesian inference in order to give readers a broad perspective. For example, the "uniformly most powerful" approach to testing is contrasted with available decision-theoretic approaches.

Inhalt

Content.- 1: Probability Models.- 1.1 Background.- 1.1.1 General Concepts.- 1.1.2 Classical Statistics.- 1.1.3 Bayesian Statistics.- 1.2 Exchangeability.- 1.2.1 Distributional Symmetry.- 1.2.2 Frequency arid Exchangeability.- 1.3 Parametric Models.- 1.3.1 Prior, Posterior, and Predictive Distributions.- 1.3.2 Improper Prior Distributions.- 1.3.3 Choosing Probability Distributions.- 1.4 DeFinetti's Representation Theorem.- 1.4.1 Understanding the Theorems.- 1.4.2 The Mathematical Statements.- 1.4.3 Some Examples.- 1.5 Proofs of DeFinetti's Theorem and Related Results.- 1.5.1 Strong Law of Large Numbers.- 1.5.2 The Bernoulli Case.- 1.5.3 The General Finite Case.- 1.5.4 The General Infinite Case.- 1.5.5 Formal Introduction to Parametric Models.- 1.6 Infinite-Dimensional Parameters.- 1.6.1 Dirichlet Processes.- 1.6.2 Tailfree Processes+.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.1.1 Notational Overview.- 2.1.2 Sufficiency.- 2.1.3 Minimal and Complete Sufficiency.- 2.1.4 Ancillarity.- 2.2 Exponential Families of Distributions.- 2.2.1 Basic Properties.- 2.2.2 Smoothness Properties.- 2.2.3 A Characterization Theorem.- 2.3 Information.- 2.3.1 Fisher Information.- 2.3.2 Kullback-Leibler Information.- 2.3.3 Conditional Information.- 2.3.4 Jeffreys' Prior.- 2.4 Extremal Families.- 2.4.1 The Main Results.- 2.4.2 Examples.- 2.4.3 Proofs+.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.1.1 Framework.- 3.1.2 Elements of Bayesian Decision Theory.- 3.1.3 Elements of Classical Decision Theory.- 3.1.4 Summary.- 3.2 Classical Decision Theory.- 3.2.1 The Role of Sufficient Statistics.- 3.2.2 Admissibility.- 3.2.3 James-Stein Estimators.- 3.2.4 Minimax Rules.- 3.2.5 Complete Classes.- 3.3 Axiomatic Derivation of Decision Theory.- 3.3.1 Definitions and Axioms.- 3.2.2 Examples.- 3.3.3 The Main Theorems.- 3.3.4 Relation to Decision Theory.- 3.3.5 Proofs of the Main Theorems.- 3.3.6 State-Dependent Utility.- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.1.1 A Special Kind of Decision Problem.- 4.1.2 Pure Significance Tests.- 4.2 Bayesian Solutions.- 4.2.1 Testing in General.- 4.2.2 Bayes Factors.- 4.3 Most Powerful Tests.- 4.3.1 Simple Hypotheses and Alternatives.- 4.3.2 Simple Hypotheses, Composite Alternatives.- 4.3.3 One-Sided Tests.- 4.3.4 Two-Sided Hypotheses.- 4.4 Unbiased Tests.- 4.4.1 General Results.- 4.4.2 Interval Hypotheses.- 4.4.3 Point Hypotheses.- 4.5 Nuisance Parameters.- 4.5.1 Neyinan Structure.- 4.5.2 Tests about Natural Parameters.- 4.5.3 Linear Combinations of Natural Parameters.- 4.5.4 Other Two-Sided Cases.- 4.5.5 Likelihood Ratio Tests.- 4.5.6 The Standard F-Test as a Bayes Rule.- 4.6 P-Values.- 4.6.1 Definitions and Examples.- 4.6.2 P-Values and Bayes Factors.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.1.1 Minimum Variance Unbiased Estimation.- 5.1.2 Lower Bounds on the Variance of Unbiased Estimators.- 5.1.3 Maximum Likelihood Estimation.- 5.1.4 Bayesian Estimation.- 5.1.5 Robust Estimation.- 5.2 Set Estimation.- 5.2.1 Confidence Sets.- 5.2.2 Prediction Sets.- 5.2.3 Tolerance Sets.- 5.2.4 Bayesian Set Estimation.- 5.2.5 Decision Theoretic Set Estimation.- 5.3 The Bootstrap.- 5.3.1 The General Concept.- 5.3.2 Standard Deviations and Bias.- 5.3.3 Bootstrap Confidence Intervals.- 5.4 Problems.- 6: Equivariance.- 6.1 Common Examples.- 6.1.1 Location Problems.- 6.1.2 Scale Problems.- 6.2 Equivariant Decision Theory.- 6.2.1 Groups of Transformations.- 6.2.2 Equivariance and Changes of Units.- 6.2.3 Minimum Risk Equivariant Decisions.- 6.3 Testing and Confidence Intervals.- 6.3.1 P-Values in Invariant Problems.- 6.3.2 Equivariant Confidence Sets.- 6.3.3 Invariant Tests.- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.1.1 Deterministic Convergence.- 7.1.2 Stochastic Convergence.- 7.1.3 The Delta Method.- 7.2 Sample Quantiles.- 7.2.1 A Single Quantile.- 7.2.2 Several Quantiles.- 7.2.3 Linear Combinations of Quantiles.- 7.3 Large Sample Estimation.- 7.3.1 Some Principles of Large Sample Estimation.- 7.3.2 Maximum Likelihood Estimators.- 7.3.3 MLEs in Exponential Families.- 7.3.4 Examples of Inconsistent MLEs.- 7.3.5 Asymptotic Normality of MLEs.- 7.3.6 Asymptotic Properties of M-Estimators.- 7.4 Large Sample Properties of Posterior Distributions.- 7.4.1 Consistency of Posterior Distributions+.- 7.4.2 Asymptotic Normality of Posterior Distributions.- 7.4.3 Laplace Approximations to Posterior Distributions.- 7.4.4 Asymptotic Agreement of Predictive Distributions+.- 7.5 Large Sample Tests.- 7.5.1 Likelihood Ratio Tests.- 7.5.2 Chi-Squarcd Goodness of Fit Tests.- 7.6 Problems.- 8: Hierarchical Models.- 8.1 Introduction.- 8.1.1 General Hierarchical Models.- 8.1.2 Partial Exchangeability.- 8.1.3 Examples of the Representation Theorem.- 8.2 Normal Linear Models.- 8.2.1 One-Way ANOVA.- 8.2.2 Two-Way Mixed Model ANOVA.- 8.2.3 Hypothesis Testing.- 8.3 Nonnormal Models.- 8.3.1 Poisson Process Data.- 8.3.2 Bernoulli Process Data.- 8.4 Empirical Bayes Analysis.- 8.4.1 Naïve Empirical Bayes.- 8.4.2 Adjusted Empirical Bayes.- 8.4.3 Unequal Variance Case.- 8.5 Successive Substitution Sampling.- 8.5.1 The General Algorithm.- 8.5.2 Normal Hierarchical Models.- 8.5.3 Nonnormal Models.- 8.6 Mixtures of Models.- 8.6.1 General Mixture Models.- 8.6.2 Outliers.- 8.6.3 Bayesian Robustness.- 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.1.1 Mathematical 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.