Bootstrap001Bootstrap002Bootstrap003Bootstrap004Bootstrap005Bootstrap006Bootstrap007Bootstrap008Bootstrap009Bootstrap010Bootstrap011)esign and Analysis of Cross-Over Trials B. JonesandM.G. Kenward (1989)15Empirical Bayes Method, 2nd edition J.S. MaritzandT.Lwin(1989)Symmetric Multivariate and Related Distributions K.-T. Fang, S. KotzandK. Ng (1989)Ieneralized Linear Models, 2nd edition P. McCullaghandJ.A. Neider (1989)38 Cyclic Designs J.A. John (1987)39 Analog Estimation Methods in Econometrics C.F. Manski (1988)40 Subset Selection in Regression A.J. Miller (1990)41 Analysis of Repeated Measures M. CrowderandD.J.Hand(1990)42 Statistical Reasoning with Imprecise ProbabilitiesP. Walley (1990)~3Generalized Additive Models T.J. HastieandR.J. Tibshirani (1990)lnspection Errors for Attributes in Quality Control N.L. Johnson, S. Kotzandx.Wu (1991)·5 The AnalysisofContingency Tables, 2nd edition B.S. Everitt (1992)46 The Analysis of Quantal Response DataB.f.T.Morgan (1992)47 Longitudinal Data with Serial Correlation: A State-Space ApproachR.H. Jones (1993): Differential Geometry and Statistics M.K. Murrayandf. W. Rice (1993)49 Markov Models and Optimization M.H.A. Davies (1993)50 Chaos and Networks: Statistical and Probabilistic Aspects Edited byO. Barndorff-Nielsen et al. (1993)Number Theoretic Methods in Statistics K.-T. FangandW. Yuan (1993)2 Inference and Asymptotics O. Barndorff-NielsenandD.R.Cox(1993);3 Practical Risk Theory for Actuaries C.D. Daykin, T. PentikainenandM. Pesonen (1993)54 Statistical Concepts and Applications in Medicine f. AitchisonandI.f.Lauder(1994)55 Predictive Inference S. Geisser (1993)56 Model-Free Curve Estimation M. TarterandM.Lock(1993)57 An Introduction to the Bootstrap B. EfronandR. Tibshirani (1993)(Full details concerning this series are available from the Publishers.)AnIntroductiontotheBootstrapBradleyEfronDepartmentofStatisticsStanfordUniversityandRobert J.TibshiraniDepartment.ofPreventative MedicineandBiostatisticsandDepartmentofStatistics, UniversityofTorontoCHAPMAN&HALL/CRCBocaRatonLondonNew York Washington, D.C.Libraryof Congress Cataloging-in-PublicationDataEfron, Bradley.An introduction to the bootstrap/Brad Efron, Rob Tibshirani.p. em.Includes bibliographical references.ISBN 0-412-04231-21. Bootstrap (Statistics).!. Tibshirani, Robert. II. Title.QA276.8.E3745 1993519.5'44-dc2093-4489CIPTOCHERYL,CHARLIE,RYANANDJULIEThisbookcontainsinformationobtained from authentic and highly regarded sources.Reprinted material is quoted with permission, and sources are indicated. Awidevariety ofreferences are listed. Reasonable efforts have been made to publish reliable data and information,but the author and the publisher cannot assume responsibility for the validityofall materials orfor the consequences of their use.Apart from any fair dealing for the purposes of research or private study, or criticism or review,as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may notbe reproduced, stored or transmitted, in any form or by any means, electronic or mechanical,including photocopying, microfihning, and recording, or by any information storage or retrievalsystem, without the prior permission in writing of the publishers, or in the case of reprographicreproduction only in accordance with the terms of the licenses issued by the Copyright LicensingAgency in the UK, or in accordance with the terms of the license issued by the appropriateReproduction Rights Organization outside the UK.The consent ofCRCPress LLC does not extend to copying for general distribution, forpromotion, for creating new works, or for resale. Specific permission must be obtained in writingfrom CRC PressLLCfor such copying.Direct all inquiries toCRCPress LLC, 2000 Corporate Blvd., N.W.,BocaRaton,Florida33431.TrademarkNotice: Product or corporate names may be trademarks or registered trademarks,and are used only for identification and explanation, without intent to infringe.First CRC Press reprint 1998Originally published byChapman& Hall© 1993 by Chapman & Hall© 1998 byCRCPress LLCNo claim to original U.S. Government worksInternational StandardBookNumber0-412-04231-2LibraryofCongress CardNumber93-4489Printed in the United States of America 2 3 4 5 6 7 8 9 0Printed on acid-freepaperANDTOTHEMEMORYOFRUPERTG.MILLER,JR.xviPREFACEsupportedthedevelopment ofstatisticaltheoryatStanford,in-cludingmuchofthetheorybehindthisbook.Thesecondauthorwould like tothankhis wifeCherylforherunderstandingandsupportduringthisentireproject,andhisparentsfor a lifetimeof encouragement. He gratefully acknowledgesthesupportoftheNaturalSciencesandEngineering Research Council ofCanada.CHAPTER1IntroductionPaloAltoandTorontoJune1993BradleyEfronRobertTibshiraniStatisticsisthescience oflearningfrom experience, especially ex-periencethatarrives alittlebitata time.Theearliestinformationscience wasstatistics,originatinginabout1650.Thiscenturyhasseenstatisticaltechniques becometheanalyticmethodsof choicein biomedical science, psychology,education,economics, communi-cationstheory, sociology, genetic studies, epidemiology,andotherareas. Recently,traditionalsciences like geology, physics,andas-tronomyhavebeguntomakeincreasing use ofstatisticalmethodsastheyfocus onareasthatdemandinformationalefficiency, such asthestudyofrareandexotic particles orextremelydistantgalaxies.Most peoplearenotnatural-bornstatisticians.Left toourowndevices wearenotverygoodatpickingoutpatternsfrom a seaof noisydata.Toputitanotherway, wearealltoogoodatpick-ingoutnon-existentpatternsthathappentosuitourpurposes.Statisticaltheoryattackstheproblemfrombothends.Itprovidesoptimalmethodsfor finding a real signal in a noisybackground,andalso providesstrictchecksagainsttheoverinterpretationofrandompatterns.Statisticaltheoryattemptsto answerthreebasic questions:(1) How should I collect mydata?(2) How should I analyzeandsummarizethedatathatI've col-lected?(3) Howaccuratearemydatasummaries?Question3constitutespartoftheprocess known asstatisticalin-ference.Thebootstrapis a recently developed technique formakingcertainkinds ofstatisticalinferences.Itis only recently developedbecause it requiresmoderncomputerpower to simplifytheoftenintricatecalculations oftraditionalstatisticaltheory.Theexplanationsthatwe will give forthebootstrap,andotherWe will seeexamplesofmuchmorecomplicatedsummariesinlaterchapters.Oneadvantageof using agoodexperimentaldesign is asimplification ofitsresults.Whatstrikestheeye here