Theory of Statistics
Published on 6. December 2012
XVI, 716 pages
978-1-4612-4250-5 (ISBN)
Description
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.
Reviews / Votes
From the reviews:
"Another excellent book in theory of statistics is by Mark J. Schervish. . Readers will enjoy reading this book to see how differently the theory can be presented . . This well written book contains nine chapters and four appendices. ... Each chapter has both easy and challenging problems. The book is suitable for graduate level statistical theory courses. Examples and illustrations are well explained. I liked the author's presentation, and learned a lot from the book. I highly recommend this book to theoretical statisticians." (Ramalingam Shanmugam, Journal of Statistical Computation and Simulation, Vol. 74 (11), November, 2004)
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Content
Content.- 1: Probability Models.- 1.1 Background.- 1.2 Exchangeability.- 1.4 DeFinetti's Representation Theorem.- 1.5 Proofs of DeFinetti's Theorem and Related Results*.- 1.6 Infinite-Dimensional Parameters*.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.2 Exponential Families of Distributions.- 2.4 Extremal Families*.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.2 Classical Decision Theory.- 3.3 Axiomatic Derivation of Decision Theory*.- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.2 Bayesian Solutions.- 4.3 Most Powerful Tests.- 4.4 Unbiased Tests.- 4.5 Nuisance Parameters.- 4.6 P-Values.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.2 Set Estimation.- 5.3 The Bootstrap*.- 5.4 Problems.- 6: Equivariance*.- 6.1 Common Examples.- 6.2 Equivariant Decision Theory.- 6.3 Testing and Confidence Intervals*.- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.2 Sample Quantiles.- 7.3 Large Sample Estimation.- 7.4 Large Sample Properties of Posterior Distributions.- 7.5 Large Sample Tests.- 7.6 Problems.- 8: Hierarchical Models.- 8.1 Introduction.- 8.3 Nonnormal Models*.- 8.4 Empirical Bayes Analysis*.- 8.5 Successive Substitution Sampling.- 8.6 Mixtures of Models.- 8.7 Problems.- 9: Sequential Analysis.- 9.1 Sequential Decision Problems.- 9.2 The Sequential Probability Ratio Test.- 9.3 Interval Estimation*.- 9.4 The Relevancc of Stopping Rules.- 9.5 Problems.- Appendix A: Measure and Integration Theory.- A.1 Overview.- A.1.1 Definitions.- A.1.2 Measurable Functions.- A.1.3 Integration.- A.1.4 Absolute Continuity.- A.2 Measures.- A.3 Measurable Functions.- A.4 Integration.- A.5 Product Spaces.- A.6 Absolute Continuity.- A.7 Problems.- Appendix B: Probability Theory.- B.1 Overview.- B.1.1Mathematical Probability.- B.1.2 Conditioning.- B.1.3 Limit Theorems.- B.2 Mathematical Probability.- B.2.1 Random Quantities and Distributions.- B.2.2 Some Useful Inequalities.- B.3 Conditioning.- B.3.1 Conditional Expectations.- B.3.2 Borel Spaces*.- B.3.3 Conditional Densities.- B.3.4 Conditional Independence.- B.3.5 The Law of Total Probability.- B.4 Limit Theorems.- B.4.1 Convergence in Distribution and in Probability.- B.4.2 Characteristic Functions.- B.5 Stochastic Processes.- B.5.1 Introduction.- B.5.3 Markov Chains*.- B.5.4 General Stochastic Processes.- B.6 Subjective Probability.- B.7 Simulation*.- B.8 Problems.- Appendix C: Mathematical Theorems Not Proven Here.- C.1 Real Analysis.- C.2 Complex Analysis.- C.3 Functional Analysis.- Appendix D: Summary of Distributions.- D.1 Univariate Continuous Distributions.- D.2 Univariate Discrete Distributions.- D.3 Multivariate Distributions.- References.- Notation and Abbreviation Index.- Name Index.
