Intended as the text for a sequence of advanced courses, this book covers major topics in theoretical statistics in a concise and rigorous fashion. The discussion assumes a background in advanced calculus, linear algebra, probability, and some analysis and topology. Measure theory is used, but the notation and basic results needed are presented in an initial chapter on probability, so prior knowledge of these topics is not essential.
The presentation is designed to expose students to as many of the central ideas and topics in the discipline as possible, balancing various approaches to inference as well as exact, numerical, and large sample methods. Moving beyond more standard material, the book includes chapters introducing bootstrap methods, nonparametric regression, equivariant estimation, empirical Bayes, and sequential design and analysis.
The book has a rich collection of exercises. Several of them illustrate how the theory developed in the book may be used in various applications. Solutions to many of the exercises are included in an appendix.
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Table of ContentsProbability and Measure.- Exponential Families.- Risk, Sufficiency, Completeness, and Ancillarity.- Unbiased Estimation.- Curved Exponential Families.- Conditional Distributions.- Bayesian Estimation.- Large-Sample Theory.- Estimating Equations and Maximum Likelihood.- Equivariant Estimation.- Empirical Bayes and Shrinkage Estimators.- Hypothesis Testing.- Optimal Tests in Higher Dimensions.- General Linear Model.- Bayesian Inference: Modeling and Computation.- Asymptotic Optimality1.- Large-Sample Theory for Likelihood Ratio Tests.- Nonparametric Regression.- Bootstrap Methods.- Sequential Methods.