Theory of Statistics [Paperback]

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.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.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 DeFinettis Representation Theorem.- 1.4.1 Understanding the Theorems.- 1.4.2 The Mathematical Statements.- 1.4.3 Some Examples.- 1.5 Proofs of DeFinettis 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 Tailfrel£,