PrefaceThis book aims to provide a concise but comprehensive account of the essential elements ofstatistical inference and theory. It is designed to be used as a text for courses on statisticaltheory for students of mathematics or statistics at the advanced undergraduate or Masterslevel (UK) or the first-year graduate level (US), or as a reference for researchers in otherfields seeking a concise treatment of the key concepts of and approaches to statisticalinference. It is intended to give a contemporary and accessible account of procedures usedto draw formal inference from data.The book focusses on a clear presentation of the main concepts and results underlyingdifferent frameworks of inference, with particular emphasis on the contrasts amongfrequentist, Fisherian and Bayesian approaches. It provides a description of basic materialon these main approaches to inference, as well as more advanced material on recentdevelopments in statistical theory, including higher-order likelihood inference, bootstrapmethods, conditional inference and predictive inference. It places particular emphasis oncontemporary computational ideas, such as applied in bootstrap methodology and Markovchain Monte Carlo techniques of Bayesian inference. Throughout, the text concentrateson concepts, rather than mathematical detail, but every effort has been made to presentthe key theoretical results in as precise and rigorous a manner as possible, consistent withthe overall mathematical level of the book. The book contains numerous extended examplesof application of contrasting inference techniques to real data, as well as selectedhistorical commentaries. Each chapter concludes with an accessible set of problems andexercises.Prerequisites for the book are calculus, linear algebra and some knowledge of basicprobability (including ideas such as conditional probability, transformations of densitiesetc., though not measure theory). Some previous familiarity with the objectives of andmain approaches to statistical inference is helpful, but not essential. Key mathematical andprobabilistic ideas are reviewed in the text where appropriate.

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Essentials of statistical inference : G.A. Young, R.L. Smith

Smith, R. L., Ross, S., Silverman, B., Young, G. A., Stein, M., Ripley, B. D., Gill, R.

Young, G. A., G. A. Young, R. L. Smith

Young, G. A., R. L. Smith, G. A. Young

G. A. Young, R. L. Smith, Young, G. A

G. A. Young, R. L. Smith, G.A YOUNG

Young, G. A., Smith, R. L.

G A Young; Richard L Smith

G. Alastair Young

Greenwich Medical Media Ltd

Cambridge series in statistical and probabilistic mathematics, Cambridge series on statistical and probabilistic mathematics, 1st pbk. ed., Cambridge, UK, England, 2010

Cambridge series in statistical and probabilistic mathematics, Cambridge series on statistical and probabilistic mathematics, Cambridge, UK, New York, England, 2005

Cambridge series on statistical and probabilistic mathematics, 16, Cambridge, UK ; New York, 2005

Cambridge series in statistical and probabilistic mathematics, First published, 2005

CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS; 16, NEW YORK, Unknown

Cambridge University Press, Cambridge, 2005

United Kingdom and Ireland, United Kingdom

Illustrated, 1, PT, 2010

July 25, 2005

lg2692276

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Includes bibliographical references (p. [218]-222) and index.

Includes bibliographical references (p. [218-222) and index.

Cover 1
Half-title 3
Series title 4
Title 5
Copyright 6
Contents 7
Preface 11
1 Introduction 13
What is statistical inference? 13
How do we approach statistical inference? 14
2 Decision theory 16
2.1 Formulation 16
2.2 The risk function 17
2.3 Criteria for a good decision rule 19
2.3.1 Admissibility 19
2.3.2 Minimax decision rules 19
2.3.3 Unbiasedness 20
2.3.4 Bayes decision rules 22
2.3.5 Some other definitions 23
2.4 Randomised decision rules 23
2.5 Finite decision problems 23
2.5.1 A story 27
2.6 Finding minimax rules in general 30
2.7 Admissibility of Bayes rules 31
2.8 Problems 31
3 Bayesian methods 34
3.1 Fundamental elements 34
3.2 The general form of Bayes rules 40
3.3 Back to minimax. . . 44
3.4 Shrinkage and the James–Stein estimator 45
3.4.1 Data example: Home run race, 1998 baseball season 47
3.4.2 Some discussion 49
3.5 Empirical Bayes 50
3.5.1 James–Stein estimator, revisited 50
3.6 Choice of prior distributions 51
3.7 Computational techniques 54
3.7.1 Gibbs sampler 54
3.7.2 The Metropolis–Hastings sampler 56
3.7.3 Metropolis–Hastings algorithm for a discrete state space 56
3.7.4 Proof of convergence of Metropolis–Hastings in the discrete case 57
3.7.5 Metropolis–Hastings algorithm for a continuous state space 58
3.7.6 Scaling 59
3.7.7 Further reading 60
3.8 Hierarchical modelling 60
3.8.1 Normal empirical Bayes model rewritten as a hierarchical model 60
3.9 Predictive distributions 64
3.9.1 An example involving baseball data 64
3.10 Data example: Coal-mining disasters 67
3.11 Data example: Gene expression data 69
3.12 Problems 72
4 Hypothesis testing 77
4.1 Formulation of the hypothesis testing problem 77
4.1.1 Test functions 78
4.1.2 Power 79
4.2 The Neyman–Pearson Theorem 80
4.3 Uniformly most powerful tests 81
4.3.1 Monotone likelihood ratio 83
4.4 Bayes factors 85
4.4.1 Bayes factors for simple hypotheses 85
4.4.2 Bayes factors for composite hypotheses 87
4.5 Problems 90
5 Special models 93
5.1 Exponential families 93
5.1.1 Definition and elementary examples 93
5.1.2 Means and variances 94
5.1.3 The natural statistics and their exact and conditional distributions 95
5.1.4 Some additional points 97
5.2 Transformation families 98
5.2.1 Maximal invariant 98
5.2.2 Equivariant statistics and a maximal invariant 99
5.2.3 An example 99
5.3 Problems 100
6 Sufficiency and completeness 102
6.1 Definitions and elementary properties 102
6.1.1 Likelihood 102
6.1.2 Sufficiency 103
6.1.3 Minimal sufficiency 103
6.1.4 Examples 105
6.2 Completeness 106
6.3 The Lehmann–Scheff ́e Theorem 107
6.4 Estimation with convex loss functions 107
6.5 Problems 108
7 Two-sided tests and conditional inference 110
7.1 Two-sided hypotheses and two-sided tests 111
7.1.1 Unbiased tests 111
7.1.2 UMPU tests for one-parameter exponential families 112
7.1.3 Testing a point null hypothesis 116
7.1.4 Some general remarks 117
7.2 Conditional inference, ancillarity and similar tests 117
7.2.1 Discussion 121
7.2.2 A more complicated example 121
7.2.3 Similar tests 123
7.2.4 Multiparameter exponential families 124
7.3 Confidence sets 126
7.3.1 Construction of confidence intervals via pivotal quantities 126
7.3.2 A general construction 128
7.3.3 Criteria for good confidence sets 128
7.4 Problems 129
8 Likelihood theory 132
8.1 Definitions and basic properties 132
8.1.1 Maximum likelihood estimator 132
8.1.2 Examples 133
8.1.3 Score function and information 134
8.1.4 Some discussion 135
8.1.5 Some mathematical reminders 136
8.2 The Cram ́er–Rao Lower Bound 137
8.3 Convergence of sequences of random variables 139
8.4 Asymptotic properties of maximum likelihood estimators 140
8.4.1 Consistency 141
8.4.2 The asymptotic distribution of the maximum likelihood estimator 141
8.4.3 Discussion 143
8.5 Likelihood ratio tests and Wilks’ Theorem 144
8.6 More on multiparameter problems 146
8.6.1 No nuisance parameter case 146
8.6.2 Nuisance parameter case: profile likelihood 147
8.6.3 Further test statistics 148
8.7 Problems 149
9 Higher-order theory 152
9.1 Preliminaries 153
9.1.1 Mann–Wald notation 153
9.1.2 Moments and cumulants 153
9.1.3 Asymptotic expansions 154
9.2 Parameter orthogonality 155
9.2.1 Definition 155
9.2.2 An immediate consequence 155
9.2.3 The case d = 1 155
9.2.4 An example 156
9.2.5 The case d > 1 156
9.2.6 Further remarks 156
9.2.7 Effects of parameter orthogonality 157
9.3 Pseudo-likelihoods 157
9.4 Parametrisation invariance 158
9.5 Edgeworth expansion 160
9.6 Saddlepoint expansion 161
9.7 Laplace approximation of integrals 164
9.8 The p formula 165
9.8.1 Introduction 165
9.8.2 Approximate ancillaries 166
9.8.3 The key formula 167
9.8.4 The adjusted signed root likelihood ratio, r 167
9.8.5 An example: Normal distribution with known coefficient of variation 169
9.8.6 The score function 170
9.9 Conditional inference in exponential families 171
9.10 Bartlett correction 172
9.11 Modified profile likelihood 173
9.12 Bayesian asymptotics 175
9.13 Problems 176
10 Predictive inference 181
10.1 Exact methods 181
10.2 Decision theory approaches 184
10.3 Methods based on predictive likelihood 187
10.3.1 Basic definitions 187
10.3.2 Butler’s predictive likelihood 189
10.3.3 Approximate predictive likelihood 190
10.3.4 Objections to predictive likelihood 191
10.4 Asymptotic methods 191
10.5 Bootstrap methods 195
10.6 Conclusions and recommendations 197
10.7 Problems 198
11 Bootstrap methods 202
11.1 An inference problem 203
11.2 The prepivoting perspective 206
11.3 Data example: Bioequivalence 213
11.4 Further numerical illustrations 215
11.5 Conditional inference and the bootstrap 220
11.6 Problems 226
Bibliography 230
Index 235

Aimed At Advanced Undergraduate And Graduate Students In Mathematics And Related Disciplines, This Book Presents The Concepts And Results Underlying The Bayesian, Frequentist And Fisherian Approaches, With Particular Emphasis On The Contrasts Between Them. Computational Ideas Are Explained, As Well As Basic Mathematical Theory. Written In A Lucid And Informal Style, This Concise Text Provides Both Basic Material On The Main Approaches To Inference, As Well As More Advanced Material On Developments In Statistical Theory, Including: Material On Bayesian Computation, Such As Mcmc, 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. G. A. Young, R. L. Smith. Title From Publisher's Bibliographic System (viewed On 01 Jun 2016). Mode Of Access: World Wide Web.

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.

"Written in an informal style, this concise text provides both basic material on the main approaches to inference, as well as more advanced material on modern developments in statistical theory, including: contemporary material on Bayesian computation, such as MCMC, 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." "Based to a large extent on lectures given at the University of Cambridge over a number of years, the material has been polished by student feedback. 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."--BOOK JACKET

This 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 basic mathematical theory as well as more advanced material, including such contemporary topics as Bayesian computation, higher-order likelihood theory, predictive inference, bootstrap methods, and conditional inference.

Aimed at advanced undergraduates and graduate students in mathematics and related disciplines, this engaging textbook gives a concise account of the main approaches to inference, with particular emphasis on the contrasts between them. It is the first textbook to synthesize contemporary material on computational topics with basic mathematical theory

<p><p>concise Account Of Main Approaches; First Textbook To Synthesize Modern Computation With Basic Theory.</p>

2020-07-26