Statistical Modelling by Exponential Families

Description

This book is a readable, digestible introduction to exponential families, encompassing statistical models based on the most useful distributions in statistical theory, such as the normal, gamma, binomial, Poisson, and negative binomial. Strongly motivated by applications, it presents the essential theory and then demonstrates the theory’s practical potential by connecting it with developments in areas such as item response analysis, social network models, conditional independence and latent variable structures, and point process models. Extensions to incomplete data models and generalized linear models are also included. In addition, the author gives a concise account of the philosophy of Per Martin-Lo¨f in order to connect statistical modelling with ideas in statistical physics, such as Boltzmann’s law. Written for graduate students and researchers with a background in basic statistical inference, the book includes a vast set of examples demonstrating models for applications and numerous exercises embedded within the text as well as at the ends of chapters. Cover......Page 1 Front Matter ......Page 2 Statistical Modelling by Exponential Families......Page 4 Copyright ......Page 5 Dedication ......Page 6 Contents ......Page 7 Examples......Page 10 Preface......Page 13 1 What Is an Exponential Family?......Page 16 2 Examples of Exponential Families......Page 21 3 Regularity Conditions and Basic Properties......Page 39 4 Asymptotic Properties of the MLE......Page 79 5 Testing Model-Reducing Hypotheses......Page 90 6 Boltzmann’s Law in Statistics......Page 115 7 Curved Exponential Families......Page 133 8 Extension to Incomplete Data......Page 158 9 Generalized Linear Models......Page 179 10 Graphical Models for Conditional Independence Structures......Page 206 11 Exponential Family Models for Graphs of Social Networks......Page 225 12 Rasch Models for Item Response and Related Model Types......Page 243 13 Models for Processes in Space or Time......Page 261 14 More Modelling Exercises......Page 273 Appendix A: Statistical Concepts and Principles......Page 280 Appendix B: Useful Mathematics......Page 283 Bibliography......Page 286 Index......Page 293