Methods of statistical model estimation

✍ Scribed by Joseph M. Hilbe, Andrew P. Robinson

• Publisher: CRC Press
• Year: 2013
• Tongue: English
• Leaves: 246
• Category: Library

✦ Synopsis

"Preface Methods of Statistical Model Estimation has been written to develop a particular pragmatic viewpoint of statistical modelling. Our goal has been to try to demonstrate the unity that underpins statistical parameter estimation for a wide range of models. We have sought to represent the techniques and tenets of statistical modelling using executable computer code. Our choice does not preclude the use of explanatory text, equations, or occasional pseudo-code. However, we have written computer code that is motivated by pedagogic considerations first and foremost. An example is in the development of a single function to compute deviance residuals in Chapter 4. We defer the details to Section 4.7, but mention here that deviance residuals are an important model diagnostic tool for GLMs. Each distribution in the exponential family has its own deviance residual, defined by the likelihood. Many statistical books will present tables of equations for computing each of these residuals. Rather than develop a unique function for each distribution, we prefer to present a single function that calls the likelihood appropriately itself. This single function replaces five or six, and in so doing, demonstrates the unity that underpins GLM. Of course, the code is less efficient and less stable than a direct representation of the equations would be, but our goal is clarity rather than speed or stability. This book also provides guidelines to enable statisticians and researchers from across disciplines to more easily program their own statistical models using R. R, more than any other statistical application, is driven by the contributions of researchers who have developed scripts, functions, and complete packages for the use of others in the general research community"--  Read more...

Programming and R Introduction R Specifics Programming Making R Packages Further Reading Statistics and Likelihood-Based Estimation Introduction Statistical Models Maximum Likelihood Estimation Interval Estimates Simulation for Fun and Profit Ordinary Regression Introduction Least-Squares Regression Maximum-Likelihood Regression Infrastructure Conclusion Generalized Linear Models Introduction GLM: Families and Terms The Exponential Family The IRLS Fitting Algorithm Bernoulli or Binary Logistic Regression Grouped Binomial Models Constructing a GLM Function GLM Negative Binomial Model Offsets Dispersion, Over and Under Goodness-of-Fit and Residual Analysis Weights Conclusion Maximum Likelihood Estimation Introduction MLE for GLM Two-Parameter MLE Panel Data What Is a Panel Model? Fixed-Effects Model Random-Intercept Model Handling More Advanced Models The EM Algorithm Further Reading Model Estimation Using Simulation Simulation: Why and When? Synthetic Statistical Models Bayesian Parameter Estimation Discussion Bibliography Index Exercises appear at the end of each chapter.

✦ Table of Contents

Front Cover
Title Page
Copyright Page
Table of Contents
Preface
Overview
Acknowledgements
1 Programming and R
1.1 Introduction
1.2 R Specifics
1.2.1 Objects
1.2.1.1 Vectors
1.2.1.2 Subsetting
1.2.2 Container Objects
1.2.2.1 Lists
1.2.2.2 Dataframes
1.2.3 Functions
1.2.3.1 Arguments
1.2.3.2 Body
1.2.3.3 Environments and Scope
1.2.4 Matrices
1.2.5 Probability Families
1.2.6 Flow Control
1.2.6.1 Conditional Execution
1.2.6.2 Loops
1.2.7 Numerical Optimization
1.3 Programming
1.3.1 Programming Style
1.3.2 Debugging
1.3.2.1 Debugging in Batch
1.3.3 Object-Oriented Programming
1.3.4 S3 Classes
1.4 Making R Packages
1.4.1 Building a Package
1.4.2 Testing
1.4.3 Installation
1.5 Further Reading
1.6 Exercises
2 Statistics and Likelihood-Based Estimation
2.1 Introduction
2.2 Statistical Models
2.3 Maximum Likelihood Estimation
2.3.1 Process
2.3.2 Estimation
2.3.2.1 Exponential Family
2.3.3 Properties
2.4 Interval Estimates
2.4.1 Wald Intervals
2.4.2 Inverting the LRT: Profile Likelihood
2.4.3 Nuisance Parameters
2.5 Simulation for Fun and Profit
2.5.1 Pseudo-Random Number Generators
2.6 Exercises
3 Ordinary Regression
3.1 Introduction
3.2 Least-Squares Regression
3.2.1 Properties
3.2.2 Matrix Representation
3.2.3 QR Decomposition
3.2.4 Example
3.3 Maximum-Likelihood Regression
3.4 Infrastructure
3.4.1 Easing Model Specification
3.4.2 Missing Data
3.4.3 Link Function
3.4.4 Initializing the Search
3.4.5 Making Failure Informative
3.4.6 Reporting Asymptotic SE and CI
3.4.7 The Regression Function
3.4.8 S3 Classes
3.4.8.1 Print
3.4.8.2 Fitted Values
3.4.8.3 Residuals
3.4.8.4 Diagnostics
3.4.8.5 Metrics of Fit
3.4.8.6 Presenting a Summary
3.4.9 Example Redux
3.4.10 Follow-up
3.5 Conclusion
3.6 Exercises
4 Generalized Linear Models
4.1 Introduction
4.2 GLM: Families and Terms
4.3 The Exponential Family
4.4 The IRLS Fitting Algorithm
4.5 Bernoulli or Binary Logistic Regression
4.5.1 IRLS
4.6 Grouped Binomial Models
4.7 Constructing a GLM Function
4.7.1 A Summary Function
4.7.2 Other Link Functions
4.8 GLM Negative Binomial Model
4.9 Offsets
4.10 Dispersion, Over- and Under-
4.11 Goodness-of-Fit and Residual Analysis
4.11.1 Goodness-of-Fit
4.11.2 Residual Analysis
4.12 Weights
4.13 Conclusion
4.14 Exercises
5 Maximum Likelihood Estimation
5.1 Introduction
5.2 MLE for GLM
5.2.1 The Log-Likelihood
5.2.2 Parameter Estimation
5.2.3 Residuals
5.2.4 Deviance
5.2.5 Initial Values
5.2.6 Printing the Object
5.2.7 GLM Function
5.2.8 Fitting for a New Family
5.3 Two-Parameter MLE
5.3.1 The Log-Likelihood
5.3.2 Parameter Estimation
5.3.3 Deviance and Deviance Residuals
5.3.4 Initial Values
5.3.5 Printing and Summarizing the Object
5.3.6 GLM Function
5.3.7 Building on the Model
5.3.8 Fitting for a New Family
5.4 Exercises
6 Panel Data
6.1 What Is a Panel Model?
6.1.1 Fixed- or Random-Effects Models
6.2 Fixed-Effects Model
6.2.1 Unconditional Fixed-Effects Models
6.2.2 Conditional Fixed-Effects Models
6.2.3 Coding a Conditional Fixed-Effects Negative Binomial
6.3 Random-Intercept Model
6.3.1 Random-Effects Models
6.3.2 Coding a Random-Intercept Gaussian Model
6.4 Handling More Advanced Models
6.5 The EM Algorithm
6.5.1 A Simple Example
6.5.2 The Random-Intercept Model
6.6 Further Reading
6.7 Exercises
7 Model Estimation Using Simulation
7.1 Simulation: Why and When?
7.2 Synthetic Statistical Models
7.2.1 Developing Synthetic Models
7.2.2 Monte Carlo Estimation
7.2.3 Reference Distributions
7.3 Bayesian Parameter Estimation
7.3.1 Gibbs Sampling
7.4 Discussion
7.5 Exercises
Bibliography
Index