Theory of statistics. (English) Zbl 0834.62002

Springer Series in Statistics. New York, NY: Springer-Verlag. xvi, 702 p. (1995).
This book contains a course of advanced statistical theory for graduate students. In contrast to most texts at this level, the book covers topics in both frequentist and Bayesian inference in great generality. The text is divided into 9 chapters.
After a brief review of elementary theory, the coverage of the subject matter begins with a detailed treatment of parametric models as motivated by DeFinetti’s representation theorem for exchangeable random variables (Chapter 1). Chapter 2 introduces sufficient statistics from both the Bayesian and non-Bayesian viewpoint. Chapter 3 is devoted to decision theory. Chapter 4 covers hypothesis testing. The contrast between the traditional uniformly most powerful approach and decision theoretical approaches is highlighted. Point and set estimation are the topics of Chapter 5. This includes unbiased and maximum likelihood estimation, confidence, prediction and tolerance sets and also robust estimation and the bootstrap. Chapter 6 is devoted to equivariant decision rules. Large sample theory is the subject of Chapter 7. Hierarchical models are presented in Chapter 8: normal linear models, nonnormal models, empirical Bayes analysis, successive substitution sampling. Some topics in sequential analysis are contained in Chapter 9.
Appendices A and B give basic definitions and theorems of measure and probability theory. Appendix C lists purely mathematical theorems that are used in the text without proof, and Appendix D gives a brief summary of the distributions used throughout the text.
Many interesting and nontrivial problems are listed at the end of every section. The presentation of the whole material is at a very high level of precision.


62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62F15 Bayesian inference
62Cxx Statistical decision theory
62A01 Foundations and philosophical topics in statistics