Methods of Statistical Model Estimation

Methods of Statistical Model Estimation
اسم المؤلف
Joseph M. Hilbe. Andrew P. Robinson
التاريخ
22 ديسمبر 2017
المشاهدات
49
التقييم
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Methods of Statistical Model Estimation
Joseph M. Hilbe
Jet Propulsion Laboratory
California Institute of Technology, USA
and Arizona State Univeristy, USA
Andrew P. Robinson
ACERA & Department of Mathematics and Statistics
The University of Melbourne, Australia
Contents
Preface ix
1 Programming and R 1
1.1 Introduction . 1
1.2 R Specifics 1
1.2.1 Objects 3
1.2.1.1 Vectors . 3
1.2.1.2 Subsetting . 7
1.2.2 Container Objects 7
1.2.2.1 Lists . 8
1.2.2.2 Dataframes . 9
1.2.3 Functions . 10
1.2.3.1 Arguments . 11
1.2.3.2 Body 13
1.2.3.3 Environments and Scope . 14
1.2.4 Matrices 16
1.2.5 Probability Families . 19
1.2.6 Flow Control . 22
1.2.6.1 Conditional Execution . 23
1.2.6.2 Loops 23
1.2.7 Numerical Optimization . 25
1.3 Programming . 27
1.3.1 Programming Style 27
1.3.2 Debugging . 28
1.3.2.1 Debugging in Batch 29
1.3.3 Object-Oriented Programming . 30
1.3.4 S3 Classes . 30
1.4 Making R Packages . 34
1.4.1 Building a Package 35
1.4.2 Testing 36
1.4.3 Installation 36
1.5 Further Reading . 37
1.6 Exercises . 37
vvi
2 Statistics and Likelihood-Based Estimation 39
2.1 Introduction . 39
2.2 Statistical Models 39
2.3 Maximum Likelihood Estimation 41
2.3.1 Process 41
2.3.2 Estimation 45
2.3.2.1 Exponential Family 46
2.3.3 Properties . 47
2.4 Interval Estimates 49
2.4.1 Wald Intervals 49
2.4.2 Inverting the LRT: Profile Likelihood . 50
2.4.3 Nuisance Parameters . 52
2.5 Simulation for Fun and Profit . 56
2.5.1 Pseudo-Random Number Generators 56
2.6 Exercises . 59
3 Ordinary Regression 61
3.1 Introduction . 61
3.2 Least-Squares Regression 62
3.2.1 Properties . 64
3.2.2 Matrix Representation 66
3.2.3 QR Decomposition 69
3.2.4 Example 71
3.3 Maximum-Likelihood Regression 74
3.4 Infrastructure 76
3.4.1 Easing Model Specification . 76
3.4.2 Missing Data . 77
3.4.3 Link Function . 78
3.4.4 Initializing the Search 78
3.4.5 Making Failure Informative . 79
3.4.6 Reporting Asymptotic SE and CI 79
3.4.7 The Regression Function . 80
3.4.8 S3 Classes . 82
3.4.8.1 Print 82
3.4.8.2 Fitted Values 83
3.4.8.3 Residuals 84
3.4.8.4 Diagnostics . 85
3.4.8.5 Metrics of Fit . 87
3.4.8.6 Presenting a Summary 89
3.4.9 Example Redux 91
3.4.10 Follow-up . 94
3.5 Conclusion 94
3.6 Exercises . 94vii
4 Generalized Linear Models 97
4.1 Introduction . 97
4.2 GLM: Families and Terms . 99
4.3 The Exponential Family . 102
4.4 The IRLS Fitting Algorithm 104
4.5 Bernoulli or Binary Logistic Regression 105
4.5.1 IRLS 111
4.6 Grouped Binomial Models . 114
4.7 Constructing a GLM Function . 120
4.7.1 A Summary Function 125
4.7.2 Other Link Functions 128
4.8 GLM Negative Binomial Model 129
4.9 Offsets 133
4.10 Dispersion, Over- and Under- 136
4.11 Goodness-of-Fit and Residual Analysis 139
4.11.1 Goodness-of-Fit 139
4.11.2 Residual Analysis . 141
4.12 Weights 143
4.13 Conclusion 143
4.14 Exercises . 144
5 Maximum Likelihood Estimation 145
5.1 Introduction . 145
5.2 MLE for GLM 146
5.2.1 The Log-Likelihood . 146
5.2.2 Parameter Estimation 148
5.2.3 Residuals . 149
5.2.4 Deviance . 150
5.2.5 Initial Values . 151
5.2.6 Printing the Object . 151
5.2.7 GLM Function 153
5.2.8 Fitting for a New Family 157
5.3 Two-Parameter MLE 160
5.3.1 The Log-Likelihood . 160
5.3.2 Parameter Estimation 162
5.3.3 Deviance and Deviance Residuals 163
5.3.4 Initial Values . 165
5.3.5 Printing and Summarizing the Object . 165
5.3.6 GLM Function 165
5.3.7 Building on the Model 171
5.3.8 Fitting for a New Family 173
5.4 Exercises . 176viii
6 Panel Data 177
6.1 What Is a Panel Model? 177
6.1.1 Fixed- or Random-Effects Models 181
6.2 Fixed-Effects Model . 181
6.2.1 Unconditional Fixed-Effects Models 181
6.2.2 Conditional Fixed-Effects Models 183
6.2.3 Coding a Conditional Fixed-Effects Negative Binomial 185
6.3 Random-Intercept Model 188
6.3.1 Random-Effects Models . 188
6.3.2 Coding a Random-Intercept Gaussian Model . 191
6.4 Handling More Advanced Models . 194
6.5 The EM Algorithm . 194
6.5.1 A Simple Example 196
6.5.2 The Random-Intercept Model 197
6.6 Further Reading . 201
6.7 Exercises . 202
7 Model Estimation Using Simulation 203
7.1 Simulation: Why and When? 203
7.2 Synthetic Statistical Models 205
7.2.1 Developing Synthetic Models 205
7.2.2 Monte Carlo Estimation . 209
7.2.3 Reference Distributions . 216
7.3 Bayesian Parameter Estimation 219
7.3.1 Gibbs Sampling 229
7.4 Discussion 230
7.5 Exercises . 231
Bibliography 233
Index 239 Pref
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