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Contents

Preface to Second Edition

Preface to First Edition

Acknowledgments

CHAPTER 1. What Is Bootstrapping?

1.1. BACKGROUND

1.2. INTRODUCTION

1.3. WIDE RANGE OF APPLICATIONS

1.4. HISTORICAL NOTES

1.5. SUMMARY

CHAPTER 2. Estimation

2.1. ESTIMATING BIAS

2.2. ESTIMATING LOCATION AND DISPERSION

2.3. HISTORICAL NOTES

CHAPTER 3. Confidence Sets and Hypothesis Testing

3.1. CONFIDENCE SETS

3.2. RELATIONSHIP BETWEEN CONFIDENCE INTERVALS AND TESTS OF HYPOTHESES

3.3. HYPOTHESIS TESTING PROBLEMS

3.4. AN APPLICATION OF BOOTSTRAP CONFIDENCE INTERVALS TO BINARY DOSE-RESPONSE MODELING

3.5. HISTORICAL NOTES

CHAPTER 4. Regression Analysis

4.1. LINEAR MODELS

4.2. NONLINEAR MODELS

4.3. NONPARAMETRIC MODELS

4.4. HISTORICAL NOTES

CHAPTER 5. Forecasting and Time Series Analysis

5.1. METHODS OF FORECASTING

5.2. TIME SERIES MODELS

5.3. WHEN DOES BOOTSTRAPPING HELP WITH PREDICTION INTERVALS?

5.4. MODEL-BASED VERSUS BLOCK RESAMPLING

5.5. EXPLOSIVE AUTOREGRESSIVE PROCESSES

5.6. BOOTSTRAPPING-STATIONARY ARMA MODELS

5.7. FREQUENCY-BASED APPROACHES

5.8. SIEVE BOOTSTRAP

5.9. HISTORICAL NOTES

CHAPTER 6. Which Resampling Method Should You Use?

6.1. RELATED METHODS

6.2. BOOTSTRAP VARIANTS

CHAPTER 7. Efficient and Effective Simulation

7.1. HOW MANY REPLICATIONS?

7.2. VARIANCE REDUCTION METHODS

7.3. WHEN CAN MONTE CARLO BE AVOIDED?

7.4. HISTORICAL NOTES

CHAPTER 8. Special Topics

8.1. SPATIAL DATA

8.2. SUBSET SELECTION

8.3. DETERMINING THE NUMBER OF DISTRIBUTIONS IN A MIXTURE MODEL

8.4. CENSORED DATA

8.5. p-VALUE ADJUSTMENT

8.6. BIOEQUIVALENCE APPLICATIONS

8.7. PROCESS CAPABILITY INDICES

8.8. MISSING DATA

8.9. POINT PROCESSES

8.10. LATTICE VARIABLES

8.11. HISTORICAL NOTES

CHAPTER 9. When Bootstrapping Fails Along with Remedies for Failures

9.1. TOO SMALL OF A SAMPLE SIZE

9.2. DISTRIBUTIONS WITH INFINITE MOMENTS

9.3. ESTIMATING EXTREME VALUES

9.4. SURVEY SAMPLING

9.5. DATA SEQUENCES THAT ARE M-DEPENDENT

9.6. UNSTABLE AUTOREGRESSIVE PROCESSES

9.7. LONG-RANGE DEPENDENCE

9.8. BOOTSTRAP DIAGNOSTICS

9.9. HISTORICAL NOTES

Bibliography 1 (Prior to 1999)

Bibliography 2 (1999-2007)

Author Index

Subject Index

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Library of Congress Cataloging-in-Publication Data:

Chernick, Michael R.
Bootstrap methods : a guide for practitioners and researchers /
Michael R. Chernick.—2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-471-75621-7 (cloth)
1. Bootstrap (Statistics) I. Title.
QA276.8.C48 2008
519.5′44—dc22
2007029309

Preface to Second Edition

Since the publication of the first edition of this book in 1999, there have been many additional and important applications in the biological sciences as well as in other fields. The major theoretical and applied books have not yet been revised. They include Hall (1992a), Efron and Tibshirani (1993), Hjorth (1994), Shao and Tu (1995), and Davison and Hinkley (1997). In addition, the bootstrap is being introduced much more often in both elementary and advanced statistics books—including Chernick and Friis (2002), which is an example of an elementary introductory biostatistics book.

The first edition stood out for (1) its use of some real-world applications not covered in other books and (2) its extensive bibliography and its emphasis on the wide variety of applications. That edition also pointed out instances where the bootstrap principle fails and why it fails. Since that time, additional modifications to the bootstrap have overcome some of the problems such as some of those involving finite populations, heavy-tailed distributions, and extreme values. Additional important references not included in the first edition are added to that bibliography. Many applied papers and other references from the period of 1999-2007 are included in a second bibliography. I did not attempt to make an exhaustive update of references.

The collection of articles entitled Frontiers in Statistics, published in 2006 by Imperial College Press as a tribute to Peter Bickel and edited by Jianqing Fan and Hira Koul, contains a section on bootstrapping and statistical learning including two chapters directly related to the bootstrap (Chapter 10, Boosting Algorithms: With an Application to Bootstrapping Multivariate Time Series; and Chapter 11, Bootstrap Methods: A Review). There is some reference to Chapter 10 from Frontiers in Statistics which is covered in the expanded Chapter 8, Special Topics; and material from Chapter 11 of Frontiers in Statistics will be used throughout the text.

Lahiri, the author of Chapter 11 in Frontiers in Statistics, has also published an excellent text on resampling methods for dependent data, Lahiri (2003a), which deals primarily with bootstrapping in dependent situations, particularly time series and spatial processes. Some of this material will be covered in Chapters 4,5,8, and 9 of this text. For time series and other dependent data, the moving block bootstrap has become the method of choice and other block bootstrap methods have been developed. Other bootstrap techniques for dependent data include transformation-based bootstrap (primarily the frequency domain bootstrap) and the sieve bootstrap. Lahiri has been one of the pioneers at developing bootstrap methods for dependent data, and his text Lahiri (2003 a) covers these methods and their statistical properties in great detail along with some results for the IID case. To my knowledge, it is the only major bootstrap text with extensive theory and applications from 2001 to 2003.

Since the first edition of my text, I have given a number of short courses on the bootstrap using materials from this and other texts as have others. In the process, new examples and illustrations have been found that are useful in a course text. The bootstrap is also being taught in many graduate school statistics classes as well as in some elementary undergraduate classes. The value of bootstrap methods is now well established.

The intention of the first edition was to provide a historical perspective to the development of the bootstrap, to provide practitioners with enough applications and references to know when and how the bootstrap can be used and to also understand its pitfalls. It had a second purpose to introduce others to the bootstrap, who may not be familiar with it, so that they can learn the basics and pursue further advances, if they are so interested. It was not intended to be used exclusively as a graduate text on the bootstrap. However, it could be used as such with supplemental materials, whereas the text by Davison and Hinkley (1997) is a self-contained graduate-level text. In a graduate course, this book could also be used as supplemental material to one of the other fine texts on bootstrap, particularly Davison and Hinkley (1997) and Efron and Tibshirani (1993). Student exercises were not included; and although the number of illustrative examples is increased in this edition, I do not include exercises at the end of the chapters.

For the most part the first edition was successful, but there were a few critics. The main complaints were with regard to lack of detail in the middle and latter chapters. There, I was sketchy in the exposition and relied on other reference articles and texts for the details. In some cases the material had too much of an encyclopedic flavor. Consequently, I have expanded on the description of the bootstrap approach to censored data in Section 8.4, and to p-value adjustment in Section 8.5. In addition to the discussion of kriging in Section 8.1,1 have added some coverage of other results for spatial data that is also covered in Lahiri (2003 a).

There are no new chapters in this edition and I tried not to add too many pages to the original bibliography, while adding substantially to Chapters 4 (on regression), 5 (on forecasting and time series), 8 (special topics), and 9 (when the bootstrap fails and remedies) and somewhat to Chapter 3 (on hypothesis testing and confidence intervals). Applications in the pharmaceutical industry such as the use of bootstrap for estimating individual and population bioequivalence are also included in a new Section 8.6.

Chapter 2 on estimating bias covered the error rate estimation problem in discriminant analysis in great detail. I find no need to expand on that material because in addition to McLachlan (1992), many new books and new editions of older books have been published on statistical pattern recognition, discriminant analysis, and machine learning that include good coverage of the bootstrap application to error rate estimation.

The first edition got mixed reviews in the technical journals. Reviews by bootstrap researchers were generally very favorable, because they recognized the value of consolidating information from diverse sources into one book. They also appreciated the objectives I set for the text and generally felt that the book met them. In a few other reviews from statisticians not very familiar with all the bootstrap applications, who were looking to learn details about the techniques, they wrote that there were too many pages devoted to the bibliography and not enough to exposition of the techniques.

My choice here is to add a second bibliography with references from 1999-2006 and early 2007. This adds about 1000 new references that I found primarily through a simple search of all articles and books with “bootstrap” as a key word or as part of the title, in the Current Index to Statistics (CIS) through my online access. For others who have access to such online searches, it is now much easier to find even obscure references as compared to what could be done in 1999 when the first edition of this book came out.

In the spirit of the first edition and in order to help readers who may not have easy access to such internet sources, I have decided to include all these new references in the second bibliography with those articles and books that are cited in the text given asterisks. This second bibliography has the citations listed in order by year of publication (starting with 1999) and in alphabetical order by first author's last name for each year. This simple addition to the bibliographies nearly doubles the size of the bibliographic section. I have also added more than a dozen references to the old bibliography [now called Bibliography 1 (prior to 1999)] from references during the period from 1985 to 1998 that were not included in the first edition.

To satisfy my critics, I have also added exposition to the chapters that needed it. I hope that I have remedied some of the criticism without sacrificing the unique aspects that some reviewers and many readers found valuable in the first edition.

I believe that in my determination to address the needs of two groups with different interests, I had to make compromises, avoiding a detailed development of theory for the first group and providing a long list of references for the second group that wanted to see the details. To better reflect and emphasize the two groups that the text is aimed at, I have changed the subtitle from A Practitioner's Guide to A Guide for Practitioners and Researchers. Also, because of the many remedies that have been devised to overcome the failures of the bootstrap and because I also include some remedies along with the failures, I have changed the title of Chapter 9 from “When does Bootstrapping Fail?” to “When Bootstrapping Fails Along with Some Remedies for Failures.”

The bibliography also was intended to help bootstrap specialists become aware of other theoretical and applied work that might appear in journals that they do not read. For them this feature may help them to be abreast of the latest advances and thus be better prepared and motivated to add to the research.

This compromise led some from the first group to feel overwhelmed by technical discussion, wishing to see more applications and not so many pages of references that they probably will never look at. For the second group, the bibliography is better appreciated but there is a desire to see more pages devoted to exposition of the theory and greater detail to the theory and more pages for applications (perhaps again preferring more pages in the text and less in the bibliography). While I did continue to expand the bibliographic section of the book, I do hope that the second edition will appeal to the critics in both groups by providing additional applications and more detailed and clear exposition of the methodology. I also hope that they will not mind the two extensive bibliographies that make my book the largest single source for extensive references on bootstrap.

Although somewhat out of date, the preface to the first edition still provides a good description of the goals of the book and how the text compares to some of its main competitors. Only objective 5 in that preface was modified. With the current state of the development of websites on the internet, it is now very easy for almost anyone to find these references online through the use of sophisticated search engines such as Yahoo's or Google's or through a CIS search.

I again invite readers to notify me of any errors or omissions in the book. There continue to be many more papers listed in the bibliographies than are referenced in the text. In order to make clear which references are cited in the text, I put an asterisk next to the cited references but I now have dispensed with a numbering according to alphabetical order, which only served to give a count of the number of books and articles cited in the text.

United BioSource Corporation
Newtown, Pennsylvania
July 2007

MICHAEL R. ChERNICK

Preface to First Edition

The bootstrap is a resampling procedure. It is named that because it involves resampling from the original data set. Some resampling procedures similar to the bootstrap go back a long way. The use of computers to do simulation goes back to the early days of computing in the late 1940s. However, it was Efron (1979a) that unified ideas and connected the simple nonparametric bootstrap, which “resamples the data with replacement,” with earlier accepted statistical tools for estimating standard errors, such as the jackknife and the delta method.

The purpose of this book is to (1) provide an introduction to the bootstrap for readers who do not have an advanced mathematical background, (2) update some of the material in the Efron and Tibshirani (1993) book by presenting results on improved confidence set estimation, estimation of error rates in discriminant analysis, and applications to a wide variety of hypothesis testing and estimation problems, (3) exhibit counterexamples to the consistency of bootstrap estimates so that the reader will be aware of the limitations of the methods, (4) connect it with some older and more traditional resampling methods including the permutation tests described by Good (1994), and (5) provide a bibliography that is extensive on the bootstrap and related methods up through 1992 with key additional references from 1993 through 1998, including new applications.

The objectives of the book are very similar to those of Davison and Hinkley (1997), especially (1) and (2). However, I differ in that this book does not contain exercises for students, but it does include a much more extensive bibliography.

This book is not a classroom text. It is intended to be a reference source for statisticians and other practitioners of statistical methods. It could be used as a supplement on an undergraduate or graduate course on resampling methods for an instructor who wants to incorporate some real-world applications and supply additional motivation for the students.

The book is aimed at an audience similar to the one addressed by Efron and Tibshirani (1993) and does not develop the theory and mathematics to the extent of Davison and Hinkley (1997). Mooney and Duval (1993) and Good (1998) are elementary accounts, but they do not provide enough development to help the practitioner gain a great deal of insight into the methods.

The spectacular success of the bootstrap in error rate estimation for discriminant functions with small training sets along with my detailed knowledge of the subject justifies the extensive coverage given to this topic in Chapter 2. A text that provides a detailed treatment of the classification problem and is the only text to include a comparison of bootstrap error rate estimates with other traditional methods is McLachlan (1992).

Mine is the first text to provide extensive coverage of real-world applications for practitioners in many diverse fields. I also provide the most detailed guide yet available to the bootstrap literature. This I hope will motivate research statisticians to make theoretical and applied advances in bootstrapping.

Several books (at least 30) deal in part with the bootstrap in specific contexts, but none of these are totally dedicated to the subject [Sprent (1998) devotes Chapter 2 to the bootstrap and provides discussion of bootstrap methods throughout his book]. Schervish (1995) provides an introductory discussion on the bootstrap in Section 5.3 and cites Young (1994) as an article that provides a good overview of the subject. Babu and Feigelson (1996) address applications of statistics in astronomy. They refer to the statistics of astronomy as astrostatistics. Chapter 5 (pp. 93-103) of the Babu-Feigelson text covers resampling methods emphasizing the bootstrap. At this point there are about a half dozen other books devoted to the bootstrap, but of these only four (Davison and Hinkley, 1997; Manly, 1997; Hjorth, 1994; Efron and Tibshirani, 1993) are not highly theoretical.

Davison and Hinkley (1997) give a good account of the wide variety of applications and provide a coherent account of the theoretical literature. They do not go into the mathematical details to the extent of Shao and Tu (1995) or Hall (1992a). Hjorth (1994) is unique in that it provides detailed coverage of model selection applications.

Although many authors are now including the bootstrap as one of the tools in a statistician’s arsenal (or for that matter in the tool kit of any practitioner of statistical methods), they deal with very specific applications and do not provide a guide to the variety of uses and the limitations of the techniques for the practitioner. This book is intended to present the practitioner with a guide to the use of the bootstrap while at the same time providing him or her with an awareness of its known current limitations. As an additional bonus, I provide an extensive guide to the research literature on the bootstrap.

This book is aimed at two audiences. The first consists of applied statisticians, engineers, scientists, and clinical researchers who need to use statistics in their work. For them, I have tried to maintain a low mathematical level. Consequently, I do not go into the details of stochastic convergence or the Edgeworth and Cornish-Fisher expansions that are important in determining the rate of convergence for various estimators and thus identify the higher-order efficiency of some of these estimators and the properties of their approximate confidence intervals.

However, I do not avoid discussion of these topics. Readers should bear with me. There is a need to understand the role of these techniques and the corresponding bootstrap theory in order to get an appreciation and understanding of how, why, and when the bootstrap works. This audience should have some background in statistical methods (at least having completed one elementary statistics course), but they need not have had courses in calculus, advanced mathematics, advanced probability, or mathematical statistics.

The second primary audience is the mathematical statistician who has done research in statistics but has not become familiar with the bootstrap but wants to learn more about it and possibly use it in future research. For him or her, my historical notes and extensive references to applications and theoretical papers will be helpful. This second audience may also appreciate the way I try to tie things together with a somewhat objective view.

To a lesser extent a third group, the serious bootstrap researcher, may find value in this book and the bibliography in particular. I do attempt to maintain technical accuracy, and the bibliography is extensive with many applied papers that may motivate further research. It is more extensive than one obtained simply by using the key word search for “bootstrap” and “resampling” in the Current Index to Statistics CD ROM. However, I would not try to claim that such a search could not uncover at least a few articles that I may have missed.

I invite readers to notify me of any errors or omissions in the book, particularly omissions regarding references. There are many more papers listed in the bibliography than are referenced in the text. In order to make clear which references are cited in the text, I put an asterisk next to the cited references along with a numbering according to alphabetical order.

Diamond Bar, California
January 1999

MICHAEL R. CHERNICK

Acknowledgments

When the first edition was written, Peter Hall was kind enough to send an advance copy of his book The Bootstrap and Edgeworth Expansion (Hall, 1992a), which was helpful to me especially in explaining the virtues of the various forms of bootstrap confidence intervals. Peter has been a major contributor to various branches of probability and statistics and has been and continues to be a major contributor to bootstrap theory and methods. I have learned a great deal about bootstrapping from Peter and his student Michael Martin, from Peter’s book, and from his many papers with Martin and others.

Brad Efron taught me mathematical statistics when I was a graduate student at Stanford. I learned about some of the early developments in bootstrapping first hand from him as he was developing his early ideas on the bootstrap. To me he was a great teacher, mentor, and later a colleague. Although I did not do my dissertation work with him and did not do research on the bootstrap until several years after my graduation, he always encouraged me and gave me excellent advice through many discussions at conferences and seminars and through our various private communications. My letters to him tended to be long and complicated. His replies to me were always brief but right to the point and very helpful. His major contributions to statistical theory include the geometry of exponential families, empirical Bayes methods, and of course the bootstrap. He also has applied the theory to numerous applications in diverse fields. Even today he is publishing important work on microarray data and applications of statistics in physics and other hard sciences. He originated the nonparametric bootstrap and developed many of its properties through the use of Monte Carlo approximations to bootstrap estimates in simulation studies. The Monte Carlo approximation provides a very practical way to use the computer to attain these estimates. Efron’s work is evident throughout this text.

This book was originally planned to be half of a two-volume series on resampling methods that Phillip Good and I started. Eventually we decided to publish separate books. Phil has since published three editions to his book, and this is the second edition of mine. Phil was very helpful to me in organizing the chapter subjects and proofreading many of my early chapters. He continually reminded me to bring out the key points first.

This book started as a bibliography that I was putting together on bootstrap in the early 1990s. The bibliography grew as I discovered, through a discussion with Brad Efron, that Joe Romano and Michael Martin also had been doing a similar thing. They graciously sent me what they had and I combined it with mine to create a large and growing bibliography that I had to continually update throughout the 1990s to keep it current and as complete as possible. Just prior to the publication of the first edition, I used the services of NERAC, a literature search firm. They found several articles that I had missed, particularly those articles that appeared in various applied journals during the period from 1993 through 1998. Gerri Beth Potash of NERAC was the key person who helped with the search. Also, Professor Robert Newcomb from the University of California at Irvine helped me search through an electronic version of the Current Index to Statistics. He and his staff at the UCI Statistical Consulting Center (especially Mira Hornbacher) were very helpful with a few other search requests that added to what I obtained from NERAC.

I am indebted to the many typists who helped produce numerous versions of the first edition. The list includes Sally Murray from Nichols Research Corporation, Cheryl Larsson from UC Irvine, and Jennifer Del Villar from Pacesetter. For the second edition I got some help learning about Latex and received guidance and encouragement from my editor Steve Quigley, Susanne Steitz and Jackie Palmieri of the Wiley editorial staff. Sue Hobson from Aux-ilium was also helpful to me in my preparation of the revised manuscript. However, the typing of the manuscript for the second edition is mine and I am responsible for any typos.

My wife Ann has been a real trooper. She helped me through my illness and allowed me the time to complete the first edition during a very busy period because my two young sons were still preschoolers. She encouraged me to finish the first edition and has been accommodating to my needs as I prepared the second. I do get the common question “Why haven’t you taken out the garbage yet?” My pat answer to that is “Later, I have to finish some work on the book first!” I must thank her for patience and perseverance.

The boys, Daniel and Nicholas, are now teenagers and are much more self-sufficient. My son Nicholas is so adept with computers now that he was able to download improved software for the word processing on my home computer.