Generalized Linear Models

Jean-François Dupuy is Professor of Applied Mathematics at the University of Rennes and is a member of the Institut de recherche mathématique de Rennes, France. His research focuses on statistical modeling, generalized linear models and duration models.

• Preface ixNotation and Acronyms xiChapter 1 Exponential Families 11.1. Definition 11.2. Mean, variance, and variance function 31.3. Examples of exponential families 41.4. Maximum likelihood estimation 91.5. Technical appendix 181.5.1. Some useful results from probability 181.5.2. Negative binomial distribution and Poisson-gamma mixtures 191.6. Exercises 20Chapter 2. From Linear Models to GLMs 252.1. Reminders on the linear model 272.1.1. Matrix form of the linear model 282.1.2. Some examples of linear models 292.1.3. Least-squares approximation 312.1.4. Asymptotics of the LSE 342.1.5. Linear Gaussian model 372.2. Three components of a generalized linear model 432.2.1. The random component 432.2.2. The linear predictor 442.2.3. The link function 452.3. Estimation in generalized linear models 462.3.1. Maximum likelihood 462.3.2. Asymptotic properties and inference 482.3.3. Estimating the dispersion parameter 502.4. Some examples 522.4.1. Logistic regression model 522.4.2. Poisson regression model 592.4.3. Gamma regression model 602.5. Generalized linear models in R: Poisson regression example 612.5.1. Confidence intervals and hypothesis tests 652.5.2. AIC and BIC, variable selection 682.5.3. Prediction, confidence intervals for a prediction 692.6. Technical appendix 712.6.1. Some probability distributions 712.6.2. Cochran’s theorem 722.7. Exercises 72Chapter 3. Censored and Missing Data in GLMs 833.1. Censored data 833.1.1. Introduction 833.1.2. Poisson regression with a right-censored response 853.1.3. Gamma regression with a right-censored response 933.2. Missing data problems 1013.2.1. Introduction 1013.2.2. Missing data typology 1023.2.3. Methods for treating missing data 1033.2.4. A missing data problem in the Poisson model 1133.2.5. A missing data problem in the gamma regression model 1273.3. Technical appendix 1353.3.1. Two lemmas 1353.3.2. Proof of theorem 3.3 1403.3.3. Proof of theorem 3.4 1433.3.4. Proof of theorem 3.5 1453.3.5. Proof of theorem 3.6 1493.3.6. Elements of empirical processes 1503.4. Exercises 154Chapter 4. Zero-Inflated Models 1594.1. Introduction 1594.1.1. Overdispersion 1594.1.2. Excess of zeros 1634.2. Zero-inflated Poisson models and extensions 1664.2.1. The zero-inflated Poisson model 1664.2.2. Semi-parametric ZIP models 1704.2.3. Zero-inflated generalized Poisson model 1744.2.4. A zero-inflation test 1774.3. Zero-inflated negative binomial model 1834.3.1. Negative binomial model 1834.3.2. ZINB model 1844.3.3. The ZIP model versus the ZINB model 1864.4. Zero-inflated binomial model 1874.5. Censored and missing data: examples of problems 1904.5.1. Censored ZIP model 1904.5.2. Missing covariables in the ZIB model 1924.5.3. Missing covariables in the ZIP model 1954.6. Marginal zero-inflated models 2004.6.1. Introduction 2004.6.2. MZIP and MZINB models 2034.7. Exercises 204References 209Index 217