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AuthorSchervish, Mark J. author
TitleTheory of Statistics [electronic resource] / by Mark J. Schervish
ImprintNew York, NY : Springer New York : Imprint: Springer, 1995
Connect tohttp://dx.doi.org/10.1007/978-1-4612-4250-5
Descript XVI, 716 p. online resource

SUMMARY

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


CONTENT

Content -- 1: Probability Models -- 1.1 Background -- 1.2 Exchangeability -- 1.4 DeFinettiโ{128}{153}s Representation Theorem -- 1.5 Proofs of DeFinettiโ{128}{153}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.1.1 Mathematical 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


Mathematics Probabilities Statistics Mathematics Probability Theory and Stochastic Processes Statistical Theory and Methods



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