As Hall (1995) notes, the idea behind the bootstrap has a long history. If we
wish to estimate a functional of a distribution function
F
, such as a population
mean
=
Z
xdF
(
x
)
;
we may use the same functional of the empirical distribution function
1
F
n
X
n
=
Z
xdF
n
(
x
)
:
The estimator so constructed is a bootstrap estimator. Yet not all functionals
can be estimated in this way. For example, a probability density cannot be so
estimated, as the empirical distribution function is pointwise discontinuous.
In 1979, Efron expanded the power of the bootstrap by showing that for
many estimation problems in which the bootstrap estimator is di¢ cult to com
pute, the estimator could be approximated by monte carlo resampling. In
essence, resamples of size
n
are drawn repeatedly from the original sample,
the estimate is calculated for each resample, and the bootstrap estimator is ap
proximated by taking an average of the appropriate function of the resample
estimates. The accuracy of the resampling approximation increases with the
number of resamples. Such a procedure provides an accurate approximation for
bootstrap estimators, such as
X
n
, which are expressed as an expectation condi
tional on the sample (or, equilivalently, as an integral with respect to
F
n
(
x
)
).
The power of resampling to provide a rich distributional approximation holds
even for small sample sizes. Consider the distribution of the sample mean with
a sample size of
n
= 7
. Because the number of possible distinct resamples is
n
n
and the number of possible distinct resample estimators of the sample mean
is
2
n
1
n
1
±
, there are 1700 di/erent possible values for the sample mean
constructed from a resample if
n
= 7
. In fact, for any smooth statistic that
does not depend on order, the bootstrap distribution of the statistic is roughly
continuous even for small sample sizes. However, for quantiles (such as the
median) things are not quite so rosy. If
n
is odd, there are only
n
distinct
values for the sample median constructed from a resample.
Because the term bootstrap estimator has become synonomous with the ap
proximate bootstrap estimator constructed from resampling, in what follows we
refer to the resampling approximation as the bootstrap estimator. We focus on
applying the bootstrap to statistics whose asymptotic distributions are indepen
dent of unknown population parameters. A statistic whose asymptotic distribu
tion is independent of unknown population parameters is called asymptotically
pivotal. Simple bootstrap procedures provide improved approximations to the
distributions of asymptotically pivotal statistics but not to the distributions
1
The empirical distribution function is the probability measure that assigns to a set a
measure equal to the proportion of sample values that lie in the set.
1
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 Fall '08
 Staff
 Econometrics, Normal Distribution, Tn, Estimation theory, Cumulative distribution function

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