Essentials of Statistical Inference

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Hardcover | 13 December 2005

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236 Pages

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25.4 x 17.78 x 1.42

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This engaging textbook presents the concepts and results underlying the Bayesian, frequentist and Fisherian approaches to statistical inference, with particular emphasis on the contrasts between them. Aimed at advanced undergraduates and graduate students in mathematics and related disciplines, it covers in a concise treatment both basic mathematical theory and more advanced material, including such contemporary topics as Bayesian computation, higher-order likelihood theory, predictive inference, bootstrap methods and conditional inference. It contains numerous extended examples of the application of formal inference techniques to real data, as well as historical commentary on the development of the subject. Throughout, the text concentrates on concepts, rather than mathematical detail, while maintaining appropriate levels of formality. Each chapter ends with a set of accessible problems. Some prior knowledge of probability is assumed, while some previous knowledge of the objectives and main approaches to statistical inference would be helpful but is not essential.

Industry Reviews

'I wish that I had such a textbook during my student days ... The text is written in a style that I sought for without too much success in my student days: sufficiently precise to satisfy a mathematician and yet omitting too much technical detail that could hide the core of the ideas. Carefully selected examples from a rainbow of application areas such as baseball, coal-mining disasters or gene expression data make it even more enjoyable to read. ... a very nice graduate level textbook ...' Journal of the Royal Statistical Society: Series A (Statistics in Society) '... gives a clear and comprehensive account of the basic elements of statistical theory ... a good text for an advanced course on statistical inference.' Publication of the International Statistical Institute 'The text presents the main concepts and results underlying different frameworks of inference, with particular emphasis on the contrasts among frequentist, Fisherian, and Bayesian approaches. It provides a depiction of basic material on these main approaches to inference, as well as more advanced material on recent developments in statistical theory, including higher-order likelihood inference, bootstrap methods, conditional inference, and predictive inference.' Zentralblatt MATH

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Preface	p. ix
Introduction	p. 1
Decision theory	p. 4
Formulation	p. 4
The risk function	p. 5
Criteria for a good decision rule	p. 7
Randomised decision rules	p. 11
Finite decision problems	p. 11
Finding minimax rules in general	p. 18
Admissibility of Bayes rules	p. 19
Problems	p. 19
Bayesian methods	p. 22
Fundamental elements	p. 22
The general form of Bayes rules	p. 28
Back to minimax…	p. 32
Shrinkage and the James-Stein estimator	p. 33
Empirical Bayes	p. 38
Choice of prior distributions	p. 39
Computational techniques	p. 42
Hierarchical modeling	p. 48
Predictive distributions	p. 52
Data example: Coal-mining disasters	p. 55
Data example: Gene expression data	p. 57
Problems	p. 60
Hypothesis testing	p. 65
Formulation of the hypothesis testing problem	p. 65
The Neyman-Pearson Theorem	p. 68
Uniformly most powerful tests	p. 69
Bayes factors	p. 73
Problems	p. 78
Special models	p. 81
Exponential families	p. 81
Transformation families	p. 86
Problems	p. 88
Sufficiency and completeness	p. 90
Definitions and elementary properties	p. 90
Completeness	p. 94
The Lehmann-Scheffé Theorem	p. 95
Estimation with convex loss functions	p. 95
Problems	p. 96
Two-sided tests and conditional inference	p. 98
Two-sided hypotheses and two-sided tests	p. 99
Conditional inference, ancillarity and similar tests	p. 105
Confidence sets	p. 114
Problems	p. 117
Likelihood theory	p. 120
Definitions and basic properties	p. 120
The Cramér-Rao Lower Bound	p. 125
Convergence of sequences of random variables	p. 127
Asymptotic properties of maximum likelihood estimators	p. 128
Likelihood ratio tests and Wilks' Theorem	p. 132
More on multiparameter problems	p. 134
Problems	p. 137
Higher-order theory	p. 140
Preliminaries	p. 141
Parameter orthogonality	p. 143
Pseudo-likelihoods	p. 145
Parametrisation invariance	p. 146
Edgeworth expansion	p. 148
Saddlepointexpansion	p. 149
Laplace approximation of integrals	p. 152
The p^* formula	p. 153
Conditional inference in exponential families	p. 159
Bartlettcorrection	p. 160
Modified profile likelihood	p. 161
Bayesian asymptotics	p. 163
Problems	p. 164
Predictive inference	p. 169
Exactmethods	p. 169
Decision theory approaches	p. 172
Methods based on predictive likelihood	p. 175
Asymptotic methods	p. 179
Bootstrap methods	p. 183
Conclusions and recommendations	p. 185
Problems	p. 186
Bootstrap methods	p. 190
An inference problem	p. 191
The prepivoting perspective	p. 194
Data example: Bioequivalence	p. 201
Further numerical illustrations	p. 203
Conditional inference and the bootstrap	p. 208
Problems	p. 214
Bibliography	p. 218
Index	p. 223
Table of Contents provided by Ingram. All Rights Reserved.

Essentials of Statistical Inference

At a Glance

Hardcover

Industry Reviews

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