An Introduction to the Bootstrap

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Lesley Poppy Francis
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1 An Introduction to the Bootstrap Bradley Efron Department of Statistics Stanford University and Robert J. Tibshirani Department of Preventative Medicine and Biostatistics and Department of Statistics, University of Zbronto CHAPMAN & HALLICRC Boca Raton London New York Washington, D.C.

2 Contents 1 Introduction An overview of this book Information for instructors Some of the notation used in the book 9 2 The accuracy of a sample mean Problems Random samples and probabilities Introduction Random samples Probability theory Problems 28 4 The empirical distribution function and the plug-in principle Introduction The empirical distribution function The plug-in principle Problems 37 5 Standard errors and estimated standard errors Introduction The standard error of a mean Estimating the standard error of.the mean Problems 43

3 Viii CONTENTS 6 The bootstrap estimate of standard error Introduction The bootstrap estimate of standard error Example: the correlation coefficient The number of bootstrap replications B The parametric bootstrap Bibliographic notes Problems 57 7 Bootstrap standard errors : some examples Introduction Example 1 : test score data Example 2: curve fitting An example of bootstrap failure Bibliographic notes Problems 82 8 More complicated data structures Introduction One-sample problems The two-sample problem More general data structures Example : lutenizing hormone The moving blocks bootstrap Bibliographic notes Problems Regression models Introduction The linear regression model Example ; the hormone data Application of the bootstrap Bootstrapping pairs vs bootstrapping residuals Example : the cell survival data Least median of squares Bibliographic notes Problems Estimates of bias 12, Introduction 124

4 1.0.2 The bootstrap estimate of bias Example: the patch data An improved estimate of bias The jackknife estimate of bias Bias correction Bibliographic notes Problems The jackknife Introduction Definition of the jackknife Example : test score data Pseudo-values Relationship between the jackknife and bootstrap Failure of the jackknife The delete-d jackknife Bibliographic notes Problems Confidence intervals based on bootstrap "tables" Introduction Some background on confidence intervals Relation between confidence intervals and hypothesis tests Student's t interval The bootstrap-t interval Transformations and the bootstrap-t Bibliographic notes Problems Confidence intervals based on bootstrap percentiles Introduction Standard normal intervals The percentile interval Is the percentile interval backwards? Coverage performance The transformation-respecting property The range-preserving property Discussion 176

5 CONTENTS 13.9 Bibliographic notes Problems Better bootstrap confidence intervals Introduction Example : the spatial test data The BCd method The ABC method Example: the tooth data Bibliographic notes Problems Permutation tests Introduction The two-sample problem Other test statistics Relationship of hypothesis tests to confidence intervals and the bootstrap Bibliographic notes Problems Hypothesis testing with the bootstrap Introduction The two-sample problem Relationship between the permutation test and the bootstrap The one-sample problem Testing multimodality of a population Discussion Bibliographic notes Problems Cross-validation and other estimates of prediction error Introduction Example: hormone data Cross-validation Cr and other estimates of prediction error Example : classification trees Bootstrap estimates o prediction error 247

6 Overview Some details The.632 bootstrap estimator Discussion Bibliographic notes Problems Adaptive estimation and calibration Introduction Example : smoothing parameter selection for curve fitting Example : calibration of a confidence point Some general considerations Bibliographic notes Problems Assessing the error in bootstrap estimates Introduction Standard error estimation Percentile estimation The jackknife-after-bootstrap Derivations Bibliographic notes Problems A geometrical representation for the bootstrap and jackknife Introduction Bootstrap sampling The jackknife as an approximation to the bootstrap Other jackknife approximations Estimates of bias An example Bibliographic notes Problems An overview of nonparametric and parametric inference Introduction Distributions, densities and likelihood functions 296

7 xii CONTENTS 21,3 FVnctional statistics and influence functions Parametric maximum likelihood inference The parametric bootstrap Relation of parametric maximum likelihood, bootstrap and jackknife approaches ,6.1 Example : influence components for the mean The empirical cdf as a maximum likelihood estimate The sandwich estimator Example: Mouse data The delta method ,9.1 Example: delta method for the mean Example : delta method for the correlation coefficient Relationship between the delta method and in finitesimal jackknife Exponential fandlies Bibliographic notes ,13 Problems Further topics in bootstrap confidence intervals Introduction Correctness and accuracy Confidence points based on approximate pivots The BC,, interval The underlying basis for the BC,, interval The ABC approximation Least favorable families The ABCq method and transformations Discussion Bibliographic notes Problems Efficient bootstrap computations Introduction Post-sampling adjustments Application to bootstrap bias estimation Application to bootstrap variance estimation Pre- and post-sampling adjustments Importance sampling for tail probabilities Application to bootstrap tail probabilities 352

8 CONTENTS xiii 23.8 Bibliographic notes Problems Approximate likelihoods Introduction Empirical likelihood Approximate pivot methods Bootstrap partial likelihood Implied likelihood Discussion Bibliographic notes Problems Bootstrap bioequivalence Introduction A bioequivalence problem Bootstrap confidence intervals Bootstrap power calculations A more careful power calculation Fieller's intervals Bibliographic notes Problems Discussion and further topics Discussion Some questions about the bootstrap References on further topics 396 Appendix : software for bootstrap computations 398 Introduction 398 Some available software 399 S language functions 399 References 413 Author index 426 Subject index 430

COPYRIGHTED MATERIAL CONTENTS

COPYRIGHTED MATERIAL CONTENTS PREFACE ACKNOWLEDGMENTS LIST OF TABLES xi xv xvii 1 INTRODUCTION 1 1.1 Historical Background 1 1.2 Definition and Relationship to the Delta Method and Other Resampling Methods 3 1.2.1 Jackknife 6 1.2.2