Contents
ix
Preface, xi
1. A Forerunner, 1
1.1 Probabilistic Inference—An Early Example, 1
References, 2
2. Frequentist Analysis, 3
2.1 Testing Using Relative Frequency, 3
2.2 Principles Guiding Frequentism, 3
2.3 Further Remarks on Tests of Significance, 5
References, 6
3. Likelihood, 7
3.1 Law of Likelihood, 7
3.2 Forms of the Likelihood Principle (LP), 11
3.3 Likelihood and Significance Testing, 13
3.4 The 2
2 Table, 14
3.5 Sampling Issues, 18
3.6 Other Principles, 21
References, 22
4. Testing Hypotheses, 25
4.1 Hypothesis Testing via the Repeated Sampling Principle, 25
4.2 Remarks on Size, 26
4.3 Uniformly Most Powerful Tests, 27
4.4 Neyman-Pearson Fundamental Lemma, 30
4.5 Monotone Likelihood Ratio Property, 37
4.6 Decision Theory, 39
4.7 Two-Sided Tests, 41
References, 43
5. Unbiased and Invariant Tests, 45
5.1 Unbiased Tests, 45
5.2 Admissibility and Tests Similar on the Boundary, 46
5.3 Neyman Structure and Completeness, 48
5.4 Invariant Tests, 55
5.5 Locally Best Tests, 62
5.6 Test Construction, 65
5.7 Remarks on N-P Theory, 68
5.8 Further Remarks on N-P Theory, 69
5.9 Law of the Iterated Logarithm (LIL), 73
5.10 Sequential Analysis, 76
5.11 Sequential Probability Ratio Test (SPRT), 76
References, 79
6. Elements of Bayesianism, 81
6.1 Bayesian Testing, 81
6.2 Testing a Composite vs. a Composite, 84
6.3 Some Remarks on Priors for the Binomial, 90
6.4 Coherence, 96
6.5 Model Selection, 101
References, 103
7. Theories of Estimation, 105
7.1 Elements of Point Estimation, 105
7.2 Point Estimation, 106
7.3 Estimation Error Bounds, 110
7.4 Efficiency and Fisher Information, 116
7.5 Interpretations of Fisher Information, 118
7.6 The Information Matrix, 122
7.7 Sufficiency, 126
7.8 The Blackwell-Rao Result, 126
7.9 Bayesian Sufficiency, 128
7.10 Maximum Likelihood Estimation, 129
7.11 Consistency of the MLE, 132
7.12 Asymptotic Normality and “Efficiency” of the MLE, 133
7.13 Sufficiency Principles, 135
References, 136
8. Set and Interval Estimation, 137
8.1 Confidence Intervals (Sets), 137
8.2 Criteria for Confidence Intervals, 139
8.3 Conditioning, 140
8.4 Bayesian Intervals (Sets), 146
8.5 Highest Probability Density (HPD) Intervals, 147
8.6 Fiducial Inference, 149
8.7 Relation Between Fiducial and Bayesian Distributions, 151
8.8 Several Parameters, 161
8.9 The Fisher-Behrens Problem, 164
8.10 Confidence Solutions, 168
8.11 The Fieller-Creasy Problem, 173
References, 182