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Theory of Statistics
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Theory of Statistics

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ISBN-13:
9780387945460
Einband:
Book
Erscheinungsdatum:
13.12.1996
Seiten:
724
Autor:
Mark J. Schervish
Gewicht:
1143 g
Format:
242x162x42 mm
Sprache:
Englisch
Beschreibung:

Aims to provide a comprehensive advanced course in the theory of statistics covering those topics in estimation, testing, and large sample theory. This book provides an account of both Classical and Bayesian inference in order to give readers a broad perspective.
'Content.- 1: Probability Models.- 1.1 Background.- 1.1.1 General Concepts.- 1.1.2 Classical Statistics.- 1.1.3 Bayesian Statistics.- 1.2 Exchangeability.- 1.2.1 Distributional Symmetry.- 1.2.2 Frequency arid Exchangeability.- 1.3 Parametric Models.- 1.3.1 Prior, Posterior, and Predictive Distributions.- 1.3.2 Improper Prior Distributions.- 1.3.3 Choosing Probability Distributions.- 1.4 DeFinetti's Representation Theorem.- 1.4.1 Understanding the Theorems.- 1.4.2 The Mathematical Statements.- 1.4.3 Some Examples.- 1.5 Proofs of DeFinetti's Theorem and Related Results .- 1.5.1 Strong Law of Large Numbers.- 1.5.2 The Bernoulli Case.- 1.5.3 The General Finite Case .- 1.5.4 The General Infinite Case.- 1.5.5 Formal Introduction to Parametric Models .- 1.6 Infinite-Dimensional Parameters .- 1.6.1 Dirichlet Processes.- 1.6.2 Tailfree Processes+.- 1.7 Problems.- 2: Sufficient Statistics.- 2.1 Definitions.- 2.1.1 Notational Overview.- 2.1.2 Sufficiency.- 2.1.3 Minimal and Complete Sufficiency.- 2.1.4 Ancillarity.- 2.2 Exponential Families of Distributions.- 2.2.1 Basic Properties.- 2.2.2 Smoothness Properties.- 2.2.3 A Characterization Theorem .- 2.3 Information.- 2.3.1 Fisher Information.- 2.3.2 Kullback-Leibler Information.- 2.3.3 Conditional Information .- 2.3.4 Jeffreys' Prior .- 2.4 Extremal Families .- 2.4.1 The Main Results.- 2.4.2 Examples.- 2.4.3 Proofs+.- 2.5 Problems.- Chapte 3: Decision Theory.- 3.1 Decision Problems.- 3.1.1 Framework.- 3.1.2 Elements of Bayesian Decision Theory.- 3.1.3 Elements of Classical Decision Theory.- 3.1.4 Summary.- 3.2 Classical Decision Theory.- 3.2.1 The Role of Sufficient Statistics.- 3.2.2 Admissibility.- 3.2.3 James-Stein Estimators.- 3.2.4 Minimax Rules.- 3.2.5 Complete Classes.- 3.3 Axiomatic Derivation of Decision Theory .- 3.3.1 Definitions and Axioms.- 3.2.2 Examples.- 3.3.3 The Main Theorems.- 3.3.4 Relation to Decision Theory.- 3.3.5 Proofs of the Main Theorems .- 3.3.6 State-Dependent Utility .- 3.4 Problems.- 4: Hypothesis Testing.- 4.1 Introduction.- 4.1.1 A Special Kind of Decision Problem.- 4.1.2 Pure Significance Tests.- 4.2 Bayesian Solutions.- 4.2.1 Testing in General.- 4.2.2 Bayes Factors.- 4.3 Most Powerful Tests.- 4.3.1 Simple Hypotheses and Alternatives.- 4.3.2 Simple Hypotheses, Composite Alternatives.- 4.3.3 One-Sided Tests.- 4.3.4 Two-Sided Hypotheses.- 4.4 Unbiased Tests.- 4.4.1 General Results.- 4.4.2 Interval Hypotheses.- 4.4.3 Point Hypotheses.- 4.5 Nuisance Parameters.- 4.5.1 Neyinan Structure.- 4.5.2 Tests about Natural Parameters.- 4.5.3 Linear Combinations of Natural Parameters.- 4.5.4 Other Two-Sided Cases .- 4.5.5 Likelihood Ratio Tests.- 4.5.6 The Standard F-Test as a Bayes Rule.- 4.6 P-Values.- 4.6.1 Definitions and Examples.- 4.6.2 P-Values and Bayes Factors.- 4.7 Problems.- 5: Estimation.- 5.1 Point Estimation.- 5.1.1 Minimum Variance Unbiased Estimation.- 5.1.2 Lower Bounds on the Variance of Unbiased Estimators.- 5.1.3 Maximum Likelihood Estimation.- 5.1.4 Bayesian Estimation.- 5.1.5 Robust Estimation .- 5.2 Set Estimation.- 5.2.1 Confidence Sets.- 5.2.2 Prediction Sets .- 5.2.3 Tolerance Sets .- 5.2.4 Bayesian Set Estimation.- 5.2.5 Decision Theoretic Set Estimation.- 5.3 The Bootstrap .- 5.3.1 The General Concept.- 5.3.2 Standard Deviations and Bias.- 5.3.3 Bootstrap Confidence Intervals.- 5.4 Problems.- 6: Equivariance .- 6.1 Common Examples.- 6.1.1 Location Problems.- 6.1.2 Scale Problems.- 6.2 Equivariant Decision Theory.- 6.2.1 Groups of Transformations.- 6.2.2 Equivariance and Changes of Units.- 6.2.3 Minimum Risk Equivariant Decisions.- 6.3 Testing and Confidence Intervals .- 6.3.1 P-Values in Invariant Problems.- 6.3.2 Equivariant Confidence Sets.- 6.3.3 Invariant Tests .- 6.4 Problems.- 7: Large Sample Theory.- 7.1 Convergence Concepts.- 7.1.1 Deterministic Convergence.- 7.1.2 Stochastic Convergence.- 7.1.3 The Delta Method.- 7.2 Sample Quantiles.-
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 account of both classical and Bayesian inference in order to give readers a broad perspective.
Commencing with chapters on probability models and the theory of sufficient statistics, the author covers decision theory, hypothesis testing, estimation, equivariance, large sample theory, hierarchical models, and, finally, sequential nalysis. Every chapter concludes with exercises which range in difficulty from the easy to the challenging. As a result, this textbook provides an excellent course in modern theoretical statistics.

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