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The Akaike information criterion (AIC) derived as an estimator of the Kullback-Leibler information discrepancy provides a useful tool for evaluating statistical models, and numerous successful applications of the AIC have been reported in various fields of natural sciences, social sciences and engineering.
One of the main objectives of this book is to provide comprehensive explanations of the concepts and derivations of the AIC and related criteria, including Schwarz's Bayesian information criterion (BIC), together with a wide range of practical examples of model selection and evaluation criteria. A secondary objective is to provide a theoretical basis for the analysis and extension of information criteria via a statistical functional approach. A generalized information criterion (GIC) and a bootstrap information criterion are presented, which provide unified tools for modeling and model evaluation for a diverse range of models, including various types of nonlinear models and model estimation procedures such as robust estimation, the maximum penalized likelihood method and a Bayesian approach.
From the Reviews: "I was fully satisfied with it. The authors are obviously well-qualified to write on the subject." (Biometrics Book Reviews, 2008) "This book explains the basic ideas of model evaluation and presents the definition and derivation of the AIC and related criteria, including BIC. ... The book makes a major contribution to the understanding of statistical modeling. Researchers interested in statistical modeling will find a lot of interesting material in it."(Erkki P. Liski, International Statistical Reviews, Vol. 76 (2), 2008) "...Modeling is an important and challenging endeavor that permeates nearly all aspects of applied statistics. The validity of inferences, predictions, and conclusions depends on the propriety of the model serving as their basis. Any book that improves the ability of practicing statisticians and biostatisticians to formulate, select and use models is worth its weight in gold. Konishi and Kitagawa have written such a book." (Journal of the American Statistical Association September 2009, Vol. 104, No. 487, Book Reviews) "With the main purpose of explaining the critical role of information criteria in statistical modeling, this book is written by two leading experts. ... The book ends with a list of references and an index. The style of writing is very good. Examples illustrate the concepts discussed and make the book immensely readable. ... Anybody interested in statistical modeling will love to read this book. ... it will be very useful to researchers and students interested in learning statistical modeling and model evaluation." (Ravi Sreenivasan, Zentralblatt MATH, Vol. 1172, 2009)
| Concept of Statistical Modeling | p. 1 |
| Role of Statistical Models | p. 1 |
| Description of Stochastic Structures by Statistical Models | p. 1 |
| Predictions by Statistical Models | p. 2 |
| Extraction of Information by Statistical Models | p. 3 |
| Constructing Statistical Models | p. 4 |
| Evaluation of Statistical Models-Road to the Information Criterion | p. 4 |
| Modeling Methodology | p. 5 |
| Organization of This Book | p. 7 |
| Statistical Models | p. 9 |
| Modeling of Probabilistic Events and Statistical Models | p. 9 |
| Probability Distribution Models | p. 10 |
| Conditional Distribution Models | p. 17 |
| Regression Models | p. 17 |
| Time Series Model | p. 24 |
| Spatial Models | p. 27 |
| Information Criterion | p. 29 |
| Kullback-Leibler Information | p. 29 |
| Definition and Properties | p. 29 |
| Examples of K-L Information | p. 32 |
| Topics on K-L Information | p. 33 |
| Expected Log-Likelihood and Corresponding Estimator | p. 35 |
| Maximum Likelihood Method and Maximum Likelihood Estimators | p. 37 |
| Log-Likelihood Function and Maximum Likelihood Estimators | p. 37 |
| Implementation of the Maximum Likelihood Method by Means of Likelihood Equations | p. 38 |
| Implementation of the Maximum Likelihood Method by Numerical Optimization | p. 40 |
| Fluctuations of the Maximum Likelihood Estimators | p. 44 |
| Asymptotic Properties of the Maximum Likelihood Estimators | p. 47 |
| Information Criterion AIC | p. 51 |
| Log-Likelihood and Expected Log-Likelihood | p. 51 |
| Necessity of Bias Correction for the Log-Likelihood | p. 52 |
| Derivation of Bias of the Log-Likelihood | p. 55 |
| Akaike Information Criterion (AIC) | p. 60 |
| Properties of MAICE | p. 69 |
| Finite Correction of the Information Criterion | p. 69 |
| Distribution of Orders Selected by AIC | p. 71 |
| Discussion | p. 73 |
| Statistical Modeling by AIC | p. 75 |
| Checking the Equality of Two Discrete Distributions | p. 75 |
| Determining the Bin Size of a Histogram | p. 77 |
| Equality of the Means and/or the Variances of Normal Distributions | p. 79 |
| Variable Selection for Regression Model | p. 84 |
| Generalized Linear Models | p. 88 |
| Selection of Order of Autoregressive Model | p. 92 |
| Detection of Structural Changes | p. 96 |
| Detection of Level Shift | p. 96 |
| Arrival Time of a Signal | p. 99 |
| Comparison of Shapes of Distributions | p. 101 |
| Selection of Box-Cox Transformations | p. 104 |
| Generalized Information Criterion (GIC) | p. 107 |
| Approach Based on Statistical Functionals | p. 107 |
| Estimators Defined in Terms of Statistical Functionals | p. 107 |
| Derivatives of the Functional and the Influence Function | p. 111 |
| Extension of the Information Criteria AIC and TIC | p. 115 |
| Generalized Information Criterion (GIC) | p. 118 |
| Definition of the GIC | p. 119 |
| Maximum Likelihood Method: Relationship Among AIC, TIC, and GIC | p. 124 |
| Robust Estimation | p. 128 |
| Maximum Penalized Likelihood Methods | p. 134 |
| Statistical Modeling by GIC | p. 139 |
| Nonlinear Regression Modeling via Basis Expansions | p. 139 |
| Basis Functions | p. 143 |
| B-Splines | p. 143 |
| Radial Basis Functions | p. 146 |
| Logistic Regression Models for Discrete Data | p. 149 |
| Linear Logistic Regression Model | p. 149 |
| Nonlinear Logistic Regression Models | p. 152 |
| Logistic Discriminant Analysis | p. 156 |
| Linear Logistic Discrimination | p. 157 |
| Nonlinear Logistic Discrimination | p. 159 |
| Penalized Least Squares Methods | p. 160 |
| Effective Number of Parameters | p. 162 |
| Theoretical Development and Asymptotic Properties of the GIC | p. 167 |
| Derivation of the GIC | p. 167 |
| Introduction | p. 167 |
| Stochastic Expansion of an Estimator | p. 170 |
| Derivation of the GIC | p. 171 |
| Asymptotic Properties and Higher-Order Bias Correction | p. 176 |
| Asymptotic Properties of Information Criteria | p. 176 |
| Higher-Order Bias Correction | p. 178 |
| Bootstrap Information Criterion | p. 187 |
| Bootstrap Method | p. 187 |
| Bootstrap Information Criterion | p. 192 |
| Bootstrap Estimation of Bias | p. 192 |
| Bootstrap Information Criterion, EIC | p. 195 |
| Variance Reduction Method | p. 195 |
| Sampling Fluctuation by the Bootstrap Method | p. 195 |
| Efficient Bootstrap Simulation | p. 196 |
| Accuracy of Bias Correction | p. 202 |
| Relation Between Bootstrap Bias Correction Terms | p. 205 |
| Applications of Bootstrap Information Criterion | p. 206 |
| Change Point Model | p. 206 |
| Subset Selection in a Regression Model | p. 208 |
| Bayesian Information Criteria | p. 211 |
| Bayesian Model Evaluation Criterion (BIC) | p. 211 |
| Definition of BIC | p. 211 |
| Laplace Approximation for Integrals | p. 213 |
| Derivation of the BIC | p. 215 |
| Extension of the BIC | p. 218 |
| Akaike's Bayesian Information Criterion (ABIC) | p. 222 |
| Bayesian Predictive Distributions | p. 224 |
| Predictive Distributions and Predictive Likelihood | p. 224 |
| Information Criterion for Bayesian Normal Linear Models | p. 226 |
| Derivation of the PIC | p. 227 |
| Numerical Example | p. 230 |
| Bayesian Predictive Distributions by Laplace Approximation | p. 231 |
| Deviance Information Criterion (DIC) | p. 236 |
| Various Model Evaluation Criteria | p. 239 |
| Cross-Validation | p. 239 |
| Prediction and Cross-Validation | p. 239 |
| Selecting a Smoothing Parameter by Cross-Validation | p. 242 |
| Generalized Cross-Validation | p. 243 |
| Asymptotic Equivalence Between AIC-Type Criteria and Cross-Validation | p. 245 |
| Final Prediction Error (FPE) | p. 247 |
| FPE | p. 247 |
| Relationship Between the AIC and FPE | p. 249 |
| Mallows' C[subscript p] | p. 251 |
| Hannan-Quinn's Criterion | p. 253 |
| ICOMP | p. 254 |
| References | p. 255 |
| Index | p. 269 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780387718866
ISBN-10: 0387718869
Series: Springer Series in Statistics
Audience:
Tertiary; University or College
Format:
Hardcover
Language:
English
Number Of Pages: 292
Published: 12th October 2007
Dimensions (cm): 23.4 x 15.6
x 1.7
Weight (kg): 0.586
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