A Leisurely Look At The Bootstrap, The Jackknife, And Cross-Validation (1983 13S) Bradley Efron.pdf

A Leisurely Look at the Bootstrap, the Jackknife, and
Cross-Validation
BRADLEY EFRON and GAlL GONG'
validation? For a quick answer. before we begin the
This is an invited expository article for The Americall main exposition. we consider a problem where none of
Statistician. It reviews the nonparametric estimation of the three methods are necessary, estimating the stan•
statistical error, mainly the bias and standard error of dard error of a sample average.
an estimator. or the error rate of a prediction rule. The The data set consists of a random sample of size II
presentation is written at a relaxed mathematical level, from an unknown probability distribution F on the real
omitting most proofs, regularity conditions. and tech• line, ...
nical details.
(l)
KEY WORDS: Bias estimation; Variance estimation: Having observed XI =Xl> X,! = X2, ... , XII = XII' •
Nonparametric standard errors; Nonparametric con• pute the sample average x = L~-1 X,lll for use as an
fidence intervals; Error rate prediction. estimate of the expectation of F.
An interesting fact. and a crucial one for statistical
applications, is that the data set provides more than the <'
estimate X. Lt also gives an estimate for the accuracy of
I. INTRODUCTION
x. namely
This article is intended to cover lots of ground, but at --
. [ I " _ III
a relaxed mathematical level that omits most proofs. cr = .. ( _ 1) 2;. (x, - x )--J : (2)
regularity conditions, and technical details. The ground Il n . ,... [
in question is the nonparamctric estimation of statistical
0- is the estimated standard error of X = X, the root
error, "Error" here refers mainly to the bias and stan•
mean squared error of estimation.
dard error of an estimator I or to the error ratc of a
The trouble with formula (2) is that it does not, in any
data-based prediction rule.
obvious way, extend to estimators other than X, for
All of the methods we discuss share some attractive
example the sample m