发帖
雷达卡
Model Selection and Model Averaging
Gerda Claeskens
K.U. Leuven
Nils Lid Hjort
University of Oslo
2008
pages 330
Contents
Preface page xi
A guide to notation xiv
1 Model selection: data examples and introduction 1
1.1 Introduction 1
1.2 Egyptian skull development 3
1.3 Who wrote ‘The Quiet Don’? 7
1.4 Survival data on primary biliary cirrhosis 10
1.5 Low birthweight data 13
1.6 Football match prediction 15
1.7 Speedskating 17
1.8 Preview of the following chapters 19
1.9 Notes on the literature 20
2 Akaike’s information criterion 22
2.1 Information criteria for balancing fit with complexity 22
2.2 Maximum likelihood and the Kullback–Leibler distance 23
2.3 AIC and the Kullback–Leibler distance 28
2.4 Examples and illustrations 32
2.5 Takeuchi’s model-robust information criterion 42
2.6 Corrected AIC for linear regression and autoregressive time series 44
2.7 AIC, corrected AIC and bootstrap-AIC for generalised
linear models∗ 46
2.8 Behaviour of AIC for moderately misspecified models∗ 49
2.9 Cross-validation 51
2.10 Outlier-robust methods 55
2.11 Notes on the literature 64
Exercises 66
vii
3 The Bayesian information criterion 70
3.1 Examples and illustrations of the BIC 70
3.2 Derivation of the BIC 78
3.3 Who wrote ‘The Quiet Don’? 82
3.4 The BIC and AIC for hazard regression models 85
3.5 The deviance information criterion 90
3.6 Minimum description length 94
3.7 Notes on the literature 96
Exercises 97
4 A comparison of some selection methods 99
4.1 Comparing selectors: consistency, efficiency and parsimony 99
4.2 Prototype example: choosing between two normal models 102
4.3 Strong consistency and the Hannan–Quinn criterion 106
4.4 Mallows’s Cp and its outlier-robust versions 107
4.5 Efficiency of a criterion 108
4.6 Efficient order selection in an autoregressive process and the FPE 110
4.7 Efficient selection of regression variables 111
4.8 Rates of convergence∗ 112
4.9 Taking the best of both worlds?∗ 113
4.10 Notes on the literature 114
Exercises 115
5 Bigger is not always better 117
5.1 Some concrete examples 117
5.2 Large-sample framework for the problem 119
5.3 A precise tolerance limit 124
5.4 Tolerance regions around parametric models 126
5.5 Computing tolerance thresholds and radii 128
5.6 How the 5000-m time influences the 10,000-m time 130
5.7 Large-sample calculus for AIC 137
5.8 Notes on the literature 140
Exercises 140
6 The focussed information criterion 145
6.1 Estimators and notation in submodels 145
6.2 The focussed information criterion, FIC 146
6.3 Limit distributions and mean squared errors in submodels 148
6.4 A bias-modified FIC 150
6.5 Calculation of the FIC 153
6.6 Illustrations and applications 154
6.7 Exact mean squared error calculations for linear regression∗ 172
6.8 The FIC for Cox proportional hazard regression models 174
6.9 Average-FIC 179
6.10 A Bayesian focussed information criterion∗ 183
6.11 Notes on the literature 188
Exercises 189
7 Frequentist and Bayesian model averaging 192
7.1 Estimators-post-selection 192
7.2 Smooth AIC, smooth BIC and smooth FIC weights 193
7.3 Distribution of model average estimators 195
7.4 What goes wrong when we ignore model selection? 199
7.5 Better confidence intervals 206
7.6 Shrinkage, ridge estimation and thresholding 211
7.7 Bayesian model averaging 216
7.8 A frequentist view of Bayesian model averaging∗ 220
7.9 Bayesian model selection with canonical normal priors∗ 222
7.10 Notes on the literature 223
Exercises 224
8 Lack-of-fit and goodness-of-fit tests 227
8.1 The principle of order selection 227
8.2 Asymptotic distribution of the order selection test 229
8.3 The probability of overfitting∗ 232
8.4 Score-based tests 236
8.5 Two or more covariates 238
8.6 Neyman’s smooth tests and generalisations 240
8.7 A comparison between AIC and the BIC for model testing∗ 242
8.8 Goodness-of-fit monitoring processes for regression models∗ 243
8.9 Notes on the literature 245
Exercises 246
9 Model selection and averaging schemes in action 248
9.1 AIC and BIC selection for Egyptian skull development data 248
9.2 Low birthweight data: FIC plots and FIC selection per stratum 252
9.3 Survival data on PBC: FIC plots and FIC selection 256
9.4 Speedskating data: averaging over covariance structure models 258
Exercises 266
10 Further topics 269
10.1 Model selection in mixed models 269
10.2 Boundary parameters 273
10.3 Finite-sample corrections∗ 281
10.4 Model selection with missing data 282
10.5 When p and q grow with n 284
10.6 Notes on the literature 285
Overview of data examples 287
References 293
Author index 306
Subject index 310
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