Bootstrap Methods: With Applications in R

Description

This book provides a compact introduction to the bootstrap method. In addition to classical results on point estimation and test theory, multivariate linear regression models and generalized linear models are covered in detail. Special attention is given to the use of bootstrap procedures to perform goodness-of-fit tests to validate model or distributional assumptions. In some cases, new methods are presented here for the first time. The text is motivated by practical examples and the implementations of the corresponding algorithms are always given directly in R in a comprehensible form. Overall, R is given great importance throughout. Each chapter includes a section of exercises and, for the more mathematically inclined readers, concludes with rigorous proofs. The intended audience is graduate students who already have a prior knowledge of probability theory and mathematical statistics. Preface Acknowledgements Contents Abbreviations 1 Introduction 1.1 Basic Idea of the Bootstrap 1.2 The R-Project for Statistical Computing 1.3 Usage of R in This Book 1.3.1 Further Non-Statistical R-Packages References 2 Generating Random Numbers 2.1 Distributions in the R-Package Stats 2.2 Uniform df. on the Unit Interval 2.3 The Quantile Transformation 2.4 The Normal Distribution 2.5 Method of Rejection 2.6 Generation of Random Vectors 2.7 Exercises References 3 The Classical Bootstrap 3.1 An Introductory Example 3.2 Basic Mathematical Background of the Classical Bootstrap 3.3 Discussion of the Asymptotic Accuracy of the Classical Bootstrap 3.4 Empirical Process and the Classical Bootstrap 3.5 Mathematical Framework of Mallow's Metric 3.6 Exercises References 4 Bootstrap-Based Tests 4.1 Introduction 4.2 The One-Sample Test 4.3 Two-Sample Tests 4.4 Goodness-of-Fit (GOF) Test 4.5 Mathematical Framework of the GOF Test 4.6 Exercises References 5 Regression Analysis 5.1 Homoscedastic Linear Regression under Fixed Design 5.1.1 Model-Based Bootstrap 5.1.2 LSE Asymptotic 5.1.3 LSE Bootstrap Asymptotic 5.2 Linear Correlation Model and the Bootstrap 5.2.1 Classical Bootstrap 5.2.2 Wild Bootstrap 5.2.3 Mathematical Framework of LSE 5.2.4 Mathematical Framework of Classical Bootstrapped LSE 5.2.5 Mathematical Framework of Wild Bootstrapped LSE 5.3 Generalized Linear Model (Parametric) 5.3.1 Mathematical Framework of MLE 5.3.2 Mathematical Framework of Bootstrap MLE 5.4 Semi-parametric Model 5.4.1 Mathematical Framework of LSE 5.4.2 Mathematical Framework of Wild Bootstrap LSE 5.5 Exercises References 6 Goodness-of-Fit Test for Generalized Linear Models 6.1 MEP in the Parametric Modeling Context 6.1.1 Implementation 6.1.2 Bike Sharing Data 6.1.3 Artificial Data 6.2 MEP in the Semi-parametric Modeling Context 6.2.1 Implementation 6.2.2 Artificial Data 6.3 Comparison of the GOF Tests under the Parametric and Semi-parametric Setup 6.4 Mathematical Framework: Marked Empirical Processes 6.4.1 The Basic MEP 6.4.2 The MEP with Estimated Model Parameters Propagating in a Fixed Direction 6.4.3 The MEP with Estimated Model Parameters Propagating in an Estimated Direction 6.5 Mathematical Framework: Bootstrap of Marked Empirical Processes 6.5.1 Bootstrap of the BMEP 6.5.2 Bootstrap of the EMEP 6.6 Exercises References Appendix A boot Package A.1 Ordinary Bootstrap A.2 Parametric Bootstrap A.3 Confidence Intervals Appendix B simTool Package Appendix C bootGOF Package Appendix D Session Info Index