Bootstrap方法详解：实践者与研究者的实用指南

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Bootstrap方法是一种统计学中的重要概念，最初由布鲁斯·艾贝尔森（Bruce Efron）在1979年提出，主要用于估计样本量有限时的参数不确定性。这个术语源自“bootstrap”，意为用自身的脚跟站立，象征着这种方法通过重复抽样来自我复制来获取可靠的结果。Bootstrap方法特别适用于那些无法简单通过大样本理论进行精确推断的情况，例如小样本、非正态分布或复杂模型下的参数估计。在《Bootstrap方法：实践者与研究人员指南》第二版中，作者Michael R. Chernick详细阐述了这一方法的原理和应用。Bootstrap方法的核心思想是通过对原始数据集进行无数次的有放回抽样（即每次抽样后将该样本放回原集合），构建出一系列与原始数据相似的“合成”样本，然后对这些样本应用所需分析，如估计均值、标准误差、置信区间等。通过比较所有这些分析结果的分布，可以得到参数估计的分布，进而得出更为稳健的误差范围。 Bootstrap方法的优势在于其简便且灵活，能够处理复杂的依赖关系和非线性问题，而且对于未知分布的假设较少。然而，它也有一定的限制，比如当数据集中存在异质性或自相关性时，结果可能不稳定。此外，Bootstrap方法的计算成本可能较高，尤其是在大规模数据集上。在实际应用中，Bootstrap方法广泛用于各种领域，包括但不限于医学研究、金融风险评估、机器学习中的模型验证、以及社会科学中的抽样误差分析。它也被视为一种强大的工具，帮助统计学家和数据科学家提高估计的精度和置信度，尤其是在数据不足或者需要深入理解样本波动性的情况下。 Bootstrap方法是一种强大的统计工具，通过模拟和估计来处理小样本或复杂问题，对于现代数据分析和模型验证具有重要意义。掌握并熟练运用Bootstrap方法，可以帮助我们在面对不确定性和复杂性时做出更为准确和可靠的决策。

xiii

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 uniﬁ ed 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 pre-

senting results on improved conﬁ dence 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 consis-

tency 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 resam-

pling 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 applica-

tions 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

xiv preface to ﬁ rst edition

the extent of Davison and Hinkley ( 1997 ). Mooney and Duval ( 1993 ) and

Good ( 1998 ) are elementary accounts, but they do not provide enough devel-

opment to help the practitioner gain a great deal of insight into the

methods.

The spectacular success of the bootstrap in error rate estimation for dis-

criminant functions with small training sets along with my detailed knowledge

of the subject justiﬁ es the extensive coverage given to this topic in Chapter 2 .

A text that provides a detailed treatment of the classiﬁ cation 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 ﬁ rst text to provide extensive coverage of real - world applica-

tions for practitioners in many diverse ﬁ elds. 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 speciﬁ c con-

texts, 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 speciﬁ c 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 ﬁ rst consists of applied statisti-

cians, 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

preface to ﬁ rst edition xv

the rate of convergence for various estimators and thus identify the higher -

order efﬁ ciency of some of these estimators and the properties of their approx-

imate conﬁ dence 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 under-

standing 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 ﬁ nd

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, particu-

larly 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 M ichael R. C hernick

January 1999

xvii

Acknowledgments

When the ﬁ rst 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 conﬁ dence intervals. Peter has been a major con-

tributor 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

ﬁ rst 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 ﬁ elds. 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,

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