Model Selection And Model Averaging

by Gerda Claeskens, Nils Lid Hjort

ISBN13: 9780521852258

ISBN10: 0521852250

Format: Hardcover

Pub. Date: 2008-07-28

Publisher(s): Cambridge University Press

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Summary

Given a data set, you can fit thousands of models at the push of a button, but how do you choose the best? With so many candidate models, overfitting is a real danger. Is the monkey who typed Hamlet actually a good writer? Choosing a model is central to all statistical work with data. We have seen rapid advances in model fitting and in the theoretical understanding of model selection, yet this book is the first to synthesize research and practice from this active field. Model choice criteria are explained, discussed and compared, including the AIC, BIC, DIC and FIC. The uncertainties involved with model selection are tackled, with discussions of frequentist and Bayesian methods; model averaging schemes are presented. Real-data examples are complemented by derivations providing deeper insight into the methodology, and instructive exercises build familiarity with the methods. The companion website features Data sets and R code.

Preface	p. xi
A guide to notation	p. xiv
Model selection: data examples and introduction	p. 1
Introduction	p. 1
Egyptian skull development	p. 3
Who wrote 'The Quiet Don'?	p. 7
Survival data on primary biliary cirrhosis	p. 10
Low birthweight data	p. 13
Football match prediction	p. 15
Speedskating	p. 17
Preview of the following chapters	p. 19
Notes on the literature	p. 20
Akaike's information criterion	p. 22
Information criteria for balancing fit with complexity	p. 22
Maximum likelihood and the Kullback-Leibler distance	p. 23
AIC and the Kullback-Leibler distance	p. 28
Examples and illustrations	p. 32
Takeuchi's model-robust information criterion	p. 43
Corrected AIC for linear regression and autoregressive time series	p. 44
AIC, corrected AIC and bootstrap-AIC for generalised linear models*	p. 46
Behaviour of AIC for moderately misspecified models*	p. 49
Cross-validation	p. 51
Outlier-robust methods	p. 55
Notes on the literature	p. 64
Exercises	p. 66
The Bayesian information criterion	p. 70
Examples and illustrations of the BIC	p. 70
Derivation of the BIC	p. 78
Who wrote 'The Quiet Don'?	p. 82
The BIC and AIC for hazard regression models	p. 85
The deviance information criterion	p. 90
Minimum description length	p. 94
Notes on the literature	p. 96
Exercises	p. 97
A comparison of some selection methods	p. 99
Comparing selectors: consistency, efficiency and parsimony	p. 99
Prototype example: choosing between two normal models	p. 102
Strong consistency and the Hannan-Quinn criterion	p. 106
Mallow's C[subscript p] and its outlier-robust versions	p. 107
Efficiency of a criterion	p. 108
Efficient order selection in an autoregressive process and the FPE	p. 110
Efficient selection of regression variables	p. 111
Rates of convergence*	p. 112
Taking the best of both worlds?*	p. 113
Notes on the literature	p. 114
Exercises	p. 115
Bigger is not always better	p. 117
Some concrete examples	p. 117
Large-sample framework for the problem	p. 119
A precise tolerance limit	p. 124
Tolerance regions around parametric models	p. 126
Computing tolerance thresholds and radii	p. 128
How the 5000-m time influences the 10,000-m time	p. 130
Large-sample calculus for AIC	p. 137
Notes on the literature	p. 140
Exercises	p. 140
The focussed information criterion	p. 145
Estimators and notation in submodels	p. 145
The focussed information criterion, FIC	p. 146
Limit distributions and mean squared errors in submodels	p. 148
A bias-modified FIC	p. 150
Calculation of the FIC	p. 153
Illustrations and applications	p. 154
Exact mean squared error calculations for linear regression*	p. 172
The FIC for Cox proportional hazard regression models	p. 174
Average-FIC	p. 179
A Bayesian focussed information criterion*	p. 183
Notes on the literature	p. 188
Exercises	p. 189
Frequentist and Bayesian model averaging	p. 192
Estimators-post-selection	p. 192
Smooth AIC, smooth BIC and smooth FIC weights	p. 193
Distribution of model average estimators	p. 195
What goes wrong when we ignore model selection?	p. 199
Better confidence intervals	p. 206
Shrinkage, ridge estimation and thresholding	p. 211
Bayesian model averaging	p. 216
A frequentist view of Bayesian model averaging*	p. 220
Bayesian model selection with canonical normal priors*	p. 223
Notes on the literature	p. 224
Exercises	p. 225
Lack-of-fit and goodness-of-fit tests	p. 227
The principle of order selection	p. 227
Asymptotic distribution of the order selection test	p. 229
The probability of overfitting*	p. 232
Score-based tests	p. 236
Two or more covariates	p. 238
Neyman's smooth tests and generalisations	p. 240
A comparison between AIC and the BIC for model testing*	p. 242
Goodness-of-fit monitoring processes for regression models*	p. 243
Notes on the literature	p. 245
Exercises	p. 246
Model selection and averaging schemes in action	p. 248
AIC and BIC selection for Egyptian skull development data	p. 248
Low birthweight data: FIC plots and FIC selection per stratum	p. 252
Survival data on PBC: FIC plots and FIC selection	p. 256
Speedskating data: averaging over covariance structure models	p. 259
Exercises	p. 266
Further topics	p. 269
Model selection in mixed models	p. 269
Boundary parameters	p. 273
Finite-sample corrections*	p. 281
Model selection with missing data	p. 282
When p and q grow with n	p. 284
Notes on the literature	p. 285
Overview of data examples	p. 287
References	p. 293
Author index	p. 306
Subject index	p. 310
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Model Selection And Model Averaging

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