Maximum Likelihood Estimation and Inference: With Examples in R, SAS and ADMB

This book takes a fresh look at the popular and well-established method of maximum likelihood for statistical estimation and inference. It begins with an intuitive introduction to the concepts and background of likelihood, and moves through to the latest developments in maximum likelihood methodology, including general latent variable models and new material for the practical implementation of integrated likelihood using the free ADMB software. Fundamental issues of statistical inference are also examined, with a presentation of some of the philosophical debates underlying the choice of statistical paradigm. Key features: Provides an accessible introduction to pragmatic maximum likelihood modelling. Covers more advanced topics, including general forms of latent variable models (including non-linear and non-normal mixed-effects and state-space models) and the use of maximum likelihood variants, such as estimating equations, conditional likelihood, restricted likelihood and integrated likelihood. Adopts a practical approach, with a focus on providing the relevant tools required by researchers and practitioners who collect and analyze real data. Presents numerous examples and case studies across a wide range of applications including medicine, biology and ecology. Features applications from a range of disciplines, with implementation in R, SAS and/or ADMB. Provides all program code and software extensions on a supporting website. Confines supporting theory to the final chapters to maintain a readable and pragmatic focus of the preceding chapters.    This book is not just an accessible and practical text about maximum likelihood, it is a comprehensive guide to modern maximum likelihood estimation and inference. It will be of interest to readers of all levels, from novice to expert. It will be of great benefit to researchers, and to students of statistics from senior undergraduate to graduate level. For use as a course text, exercises are provided at the end of each chapter.

Table of Contents:
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(θ) 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 θ  RI 236 11.3 Cramer-Rao inequality for functions of θ 239 11.4 Alternative formulae for I (θ) 241 11.5 The iid data case 243 11.6 The multi-dimensional case, θ  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