Theory of statistics
type: Item
instance of: scholarly article
title: Theory of statistics
zbMATH Open document ID: 0834.62002
author: M. J. Schervish
published in: Springer Series in Statistics
publication date: 1995-10-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62-01
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62Cxx
Mathematics Subject Classification ID: 62A01
zbMATH DE Number: 802339
zbMATH Keywords: frequentist inference
zbMATH Keywords: large sample theory
zbMATH Keywords: hierarchical models
zbMATH Keywords: advanced statistical theory
zbMATH Keywords: parametric models
zbMATH Keywords: DeFinetti's representation theorem
zbMATH Keywords: exchangeable random variables
zbMATH Keywords: sufficient statistics
zbMATH Keywords: hypothesis testing
zbMATH Keywords: maximum likelihood estimation
zbMATH Keywords: confidence
zbMATH Keywords: prediction
zbMATH Keywords: tolerance sets
zbMATH Keywords: robust estimation
zbMATH Keywords: bootstrap
zbMATH Keywords: equivariant decision rules
zbMATH Keywords: normal linear models
zbMATH Keywords: nonnormal models
zbMATH Keywords: empirical Bayes analysis
zbMATH Keywords: successive substitution sampling
zbMATH Keywords: sequential analysis
reviewed by: Jarosław Bartoszewicz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 802339