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**Statistical decision theory. Estimation, testing, and selection.**
*(English)*
Zbl 1154.62008

Springer Series in Statistics. New York, NY: Springer (ISBN 978-0-387-73193-3/hbk). xvii, 677 p. (2008).

This monograph is written not only as a basis for graduate courses, but also as a reference tool. Readers should be familiar with basic concepts of probability theory, mathematical statistics, and analysis. Additional information is presented in condensed form in an appendix. It book acquaints readers with the concepts of classical finite sample size decision theory and modern asymptotic decision theory in the sense of LeCam. Systematic applications to the fields of parameter estimation, hypotheses testing, and selection of populations are included. The central theme is what optimal decisions are in general and in specific decision problems, and how to derive them. Optimality is understood in terms of the expected loss, i.e., the risk, or some functional of it.

The fundamental probabilistic concepts and technical tools are provided in chapter 1. The central topic of chapter 2 is the Neyman-Pearson lemma and its extensions. An introduction to the general framework of decision theory is given in the next chapter. Chapter 4 is on comparison of models and reduction by sufficiency. Invariant statistical decision models and large sample approximations of models and decisions are the topics of chapters 5 and 6. In regard of optimality, estimators, tests, and selection rules are initially considered in the book side by side. But the chapters 7 to 9 deal with them individually. Many problems are presented throughout the text. Solutions to selected problems are presented at the end of each chapter.

This book is unique in offering a fuller point of view of selection rules, and the last chapter provides a thorough presentation of optimal selection rule theory. An other feature is that it combines innovation and tradition. This monograph uniquely synthesizes otherwise disparate materials, while establishing connections between classical and modern decision theory. Actually it creates a bridge between the classical results of mathematical statistics and the modern asymptotic decision theory founded by LeCam. This book also provides a broad coverage of both the frequentist and the Bayes approaches in decision theory. The Bayes approach is considered to be a useful decision-theoretic framework among others, and it is used heavily throughout the book; however, this is done without extra nonmathematical philosophical justification.

The fundamental probabilistic concepts and technical tools are provided in chapter 1. The central topic of chapter 2 is the Neyman-Pearson lemma and its extensions. An introduction to the general framework of decision theory is given in the next chapter. Chapter 4 is on comparison of models and reduction by sufficiency. Invariant statistical decision models and large sample approximations of models and decisions are the topics of chapters 5 and 6. In regard of optimality, estimators, tests, and selection rules are initially considered in the book side by side. But the chapters 7 to 9 deal with them individually. Many problems are presented throughout the text. Solutions to selected problems are presented at the end of each chapter.

This book is unique in offering a fuller point of view of selection rules, and the last chapter provides a thorough presentation of optimal selection rule theory. An other feature is that it combines innovation and tradition. This monograph uniquely synthesizes otherwise disparate materials, while establishing connections between classical and modern decision theory. Actually it creates a bridge between the classical results of mathematical statistics and the modern asymptotic decision theory founded by LeCam. This book also provides a broad coverage of both the frequentist and the Bayes approaches in decision theory. The Bayes approach is considered to be a useful decision-theoretic framework among others, and it is used heavily throughout the book; however, this is done without extra nonmathematical philosophical justification.

Reviewer: R. Schlittgen (Hamburg)