Principles of Statistical Inference

In this important book, D. R. Cox develops the key concepts of the theory of statistical inference, in particular describing and comparing the main ideas and controversies over foundational issues that have rumbled on for more than 200 years. Continuing a 60-year career of contribution to statistical thought, Professor Cox is ideally placed to give the comprehensive, balanced account of the field that is now needed.

List of examples

ix

Preface

xiii

1 Preliminaries

1

(16)

Summary

1

(1)

1.1 Starting point

1

(2)

1.2 Role of formal theory of inference

3

(1)

1.3 Some simple models

3

(4)

1.4 Formulation of objectives

7

(1)

1.5 Two broad approaches to statistical inference

7

(3)

1.6 Some further discussion

10

(3)

1.7 Parameters

13

(1)

Notes 1

14

(3)

2 Some concepts and simple applications

17

(13)

Summary

17

(1)

2.1 Likelihood

17

(1)

2.2 Sufficiency

18

(2)

2.3 Exponential family

20

(3)

2.4 Choice of priors for exponential family problems

23

(1)

2.5 Simple frequentist discussion

24

(1)

2.6 Pivots

25

(2)

Notes 2

27

(3)

3 Significance tests

30

(15)

Summary

30

(1)

3.1 General remarks

30

(1)

3.2 Simple significance test

31

(4)

3.3 One- and two-sided tests

35

(1)

3.4 Relation with acceptance and rejection

36

(1)

3.5 Formulation of alternatives and test statistics

36

(4)

3.6 Relation with interval estimation

40

(1)

3.7 Interpretation of significance tests

41

(1)

3.8 Bayesian testing

42

(1)

Notes 3

43

(2)

4 More complicated situations

45

(19)

Summary

45

(1)

4.1 General remarks

45

(1)

4.2 General Bayesian formulation

45

(2)

4.3 Frequentist analysis

47

(3)

4.4 Some more general frequentist developments

50

(9)

4.5 Some further Bayesian examples

59

(3)

Notes 4

62

(2)

5 Interpretations of uncertainty

64

(32)

Summary

64

(1)

5.1 General remarks

64

(1)

5.2 Broad roles of probability

65

(1)

5.3 Frequentist interpretation of upper limits

66

(2)

5.4 Neyman—Pearson operational criteria

68

(1)

5.5 Some general aspects of the frequentist approach

68

(1)

5.6 Yet more on the frequentist approach

69

(2)

5.7 Personalistic probability

71

(2)

5.8 Impersonal degree of belief

73

(3)

5.9 Reference priors

76

(2)

5.10 Temporal coherency

78

(1)

5.11 Degree of belief and frequency

79

(1)

5.12 Statistical implementation of Bayesian analysis

79

(5)

5.13 Model uncertainty

84

(1)

5.14 Consistency of data and prior

85

(1)

5.15 Relevance of frequentist assessment

85

(3)

5.16 Sequential stopping

88

(3)

5.17 A simple classification problem

91

(2)

Notes 5

93

(3)

6 Asymptotic theory

96

(37)

Summary

96

(1)

6.1 General remarks

96

(1)

6.2 Scalar parameter

97

(10)

6.3 Multidimensional parameter

107

(2)

6.4 Nuisance parameters

109

(5)

6.5 Tests and model reduction

114

(3)

6.6 Comparative discussion

117

(2)

6.7 Profile likelihood as an information summarizer

119

(1)

6.8 Constrained estimation

120

(4)

6.9 Semi-asymptotic arguments

124

(1)

6.10 Numerical-analytic aspects

125

(3)

6.11 Higher-order asymptotics

128

(2)

Notes 6

130

(3)

7 Further aspects of maximum likelihood

133

(28)

Summary

133

(1)

7.1 Multimodal likelihoods

133

(2)

7.2 Irregular form

135

(4)

7.3 Singular information matrix

139

(2)

7.4 Failure of model

141

(1)

7.5 Unusual parameter space

142

(2)

7.6 Modified likelihoods

144

(15)

Notes 7

159

(2)

8 Additional objectives

161

(17)

Summary

161

(1)

8.1 Prediction

161

(1)

8.2 Decision analysis

162

(1)

8.3 Point estimation

163

(6)

8.4 Non-likelihood-based methods

169

(6)

Notes 8

175

(3)

9 Randomization-based analysis

178

(16)

Summary

178

(1)

9.1 General remarks

178

(1)

9.2 Sampling a finite population

179

(5)

9.3 Design of experiments

184

(8)

Notes 9

192

(2)

Appendix A: A brief history

194

(3)

Appendix B: A personal view

197

(4)

References

201

(8)

Author index

209

(4)

Subject index

213

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Principles of Statistical Inference

Summary

Table of Contents

Supplemental Materials