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

Wiley, 2011. — 366 p. — ISBN: 0470094826.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 featuresProvides 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.Preliminaries.
A taste of likelihood.Motivating example.
Using SAS, R and ADMB.
Implementation of the motivating example.
Exercises.
Essential concepts and iid examples.Some necessary notation.
Interpretation of likelihood.
IID examples.
Exercises.Pragmatics.
Hypothesis tests and confidence intervals or regions.Approximate normality of MLEs.
Wald tests, confidence intervals and regions.
Likelihood ratio tests, confidence intervals and regions.
Likelihood ratio examples.
Profile likelihood.
Exercises.
What you really need to know.Inference about g(θ).
Wald statistics – quick and dirty?
Model selection.
Bootstrapping.
Prediction.
Things that can mess you up.
Exercises.
Maximizing the likelihood.The Newton-Raphson algorithm.
The EM (Expectation–Maximization) algorithm.
Multi-stage maximization.
Exercises.
Some widely used applications of maximum likelihood.Box-Cox transformations.
Models for survival-time data.
Mark–recapture models.
Exercises.
Generalized linear models and extensions.Specification of a GLM.
Likelihood calculations.
Model evaluation.
Case study: Logistic regression and inverse prediction in R.
Beyond binomial and Poisson models.
Case study : Multiplicative vs additive models of over-dispersed counts in SAS.
Exercises.
Quasi-likelihood and generalized estimating equations.Wedderburn’s quasi-likelihood.
Generalized estimating equations.
Exercises.
ML inference in the presence of incidental parameters.Conditional likelihood.
Integrated likelihood.
Justification.
Uses of integrated likelihood.
Exercises.
Latent variable models.Developing the likelihood.
Software.
One-way linear random-effects model.
Nonlinear mixed-effects model.
Generalized linear mixed-effects model.
State-space model for count data.
ADMB template files.
Exercises.Theoretical foundations.
Cramer-Rao inequality and Fisher information.The Cramer-Rao inequality for θ RI.
Cramer-Rao inequality for functions of θ.
Alternative formulae for I (θ).
The IID data case.
The multi-dimensional case, θ RI s.
Examples of Fisher information calculation.
Exercises.
Asymptotic theory and approximate normality.Consistency and asymptotic normality.
Approximate normality.
Wald tests and confidence regions.
Likelihood ratio test statistic.
Rao-score test statistic.
Exercises.
Tools of the trade.Equivalence of tests and confidence intervals.
Transformation of variables.
Mean and variance conditional identities.
Relevant inequalities.
Asymptotic probability theory.
Exercises.
Fundamental paradigms and principles of inference.Sufficiency principle.
Conditionality principle.
The likelihood principle.
Statistical significance versus statistical evidence.
Exercises.
Miscellanea.
Notation.
Acronyms.
Do you think like a frequentist or a Bayesian?
Some useful distributions.
Software extras.
Automatic differentiation.Appendix: Partial solutions to selected exercises.