Preface xiii
Part I PRELIMINARIES 1
1 A taste of likelihood 3
1.1 Introduction 3
1.2 Motivating example 4
1.3 Using SAS, R and ADMB 9
1.4 Implementation of the motivating example 11
1.5 Exercises 17
2 Essential concepts and iid examples 18
2.1 Introduction 18
2.2 Some necessary notation 19
2.3 Interpretation of likelihood 23
2.4 IID examples 25
2.5 Exercises 33
Part II PRAGMATICS 37
3 Hypothesis tests and confidence intervals or regions
39
3.1 Introduction 39
3.2 Approximate normality of MLEs 40
3.3 Wald tests, confidence intervals and regions 43
3.4 Likelihood ratio tests, confidence intervals and regions
49
3.5 Likelihood ratio examples 54
3.6 Profile likelihood 57
3.7 Exercises 59
4 What you really need to know 64
4.1 Introduction 64
4.2 Inference about g(?) 65
4.3 Wald statistics ? quick and dirty? 75
4.4 Model selection 79
4.5 Bootstrapping 81
4.6 Prediction 91
4.7 Things that can mess you up 95
4.8 Exercises 98
5 Maximizing the likelihood 101
5.1 Introduction 101
5.2 The Newton-Raphson algorithm 103
5.3 The EM (Expectation?Maximization) algorithm 104
5.4 Multi-stage maximization 113
5.5 Exercises 118
6 Some widely used applications of maximum likelihood
121
6.1 Introduction 121
6.2 Box-Cox transformations 122
6.3 Models for survival-time data 125
6.4 Mark?recapture models 134
6.5 Exercises 141
7 Generalized linear models and extensions 143
7.1 Introduction 143
7.2 Specification of a GLM 144
7.3 Likelihood calculations 148
7.4 Model evaluation 149
7.5 Case study 1: Logistic regression and inverse prediction in
R 154
7.6 Beyond binomial and Poisson models 161
7.7 Case study 2: Multiplicative vs additive models of
over-dispersed counts in SAS 167
7.8 Exercises 173
8 Quasi-likelihood and generalized estimating equations
175
8.1 Introduction 175
8.2 Wedderburn?s quasi-likelihood 177
8.3 Generalized estimating equations 181
8.4 Exercises 187
9 ML inference in the presence of incidental parameters
188
9.1 Introduction 188
9.2 Conditional likelihood 192
9.3 Integrated likelihood 198
9.3.1 Justification 199
9.3.2 Uses of integrated likelihood 200
9.4 Exercises 201
10 Latent variable models 202
10.1 Introduction 202
10.2 Developing the likelihood 203
10.3 Software 204
10.4 One-way linear random-effects model 210
10.5 Nonlinear mixed-effects model 217
10.6 Generalized linear mixed-effects model 221
10.7 State-space model for count data 227
10.8 ADMB template files 228
10.9 Exercises 232
Part III THEORETICAL FOUNDATIONS 233
11 Cramer-Rao inequality and Fisher information 235
11.1 Introduction 235
11.2 The Cramer-Rao inequality for ? RI
236
11.3 Cramer-Rao inequality for functions of ?
239
11.4 Alternative formulae for I (?)
241
11.5 The iid data case 243
11.6 The multi-dimensional case, ? RI
s 243
11.7 Examples of Fisher information calculation 247
11.8 Exercises 253
12 Asymptotic theory and approximate normality 256
12.1 Introduction 256
12.2 Consistency and asymptotic normality 257
12.3 Approximate normality 271
12.4 Wald tests and confidence regions 276
12.5 Likelihood ratio test statistic 280
12.6 Rao-score test statistic 281
12.7 Exercises 283
13 Tools of the trade 286
13.1 Introduction 286
13.2 Equivalence of tests and confidence intervals 286
13.3 Transformation of variables 287
13.4 Mean and variance conditional identities 288
13.5 Relevant inequalities 289
13.6 Asymptotic probability theory 291
13.7 Exercises 297
14 Fundamental paradigms and principles of inference
299
14.1 Introduction 299
14.2 Sufficiency principle 300
14.3 Conditionality principle 304
14.4 The likelihood principle 306
14.5 Statistical significance versus statistical evidence
309
14.6 Exercises 311
15 Miscellanea 313
15.1 Notation 313
15.2 Acronyms 315
15.3 Do you think like a frequentist or a Bayesian? 315
15.4 Some useful distributions 316
15.5 Software extras 321
15.6 Automatic differentiation 323
Appendix: Partial solutions to selected exercises 325
Bibliography 337
Index 345
