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We can improve on the interval estimate {142.25, 158.25} if we are
willing to accept a small probability that the interval will fail to include
the true value of the population median. We will take several hundred
bootstrap samples instead of a mere 50, and we will use the 5th and
95th percentiles of the resulting bootstrap distribution to establish the
boundaries of a 90% confidence interval.
This method might be used equally well to obtain an interval estimate
for any other population attribute: the mean and variance, the 5th per-
centile or the 25th, and the interquartile range. When several observations
are made simultaneously on each subject, the bootstrap can be used to
estimate covariances and correlations among the variables. The bootstrap is
particularly valuable when trying to obtain an interval estimate for a ratio
or for the mean and variance of a nonsymmetric distribution.
Unfortunately, such intervals have two deficiencies:
1. They are biased; that is, they are more likely to contain certain
false values of the parameter being estimated than the true one
(Efron, 1987).
2. They are wider and less efficient than they could be (Efron,
1987).
Two methods have been proposed to correct these deficiencies; let us
consider each in turn.
The first is the Hall-Wilson [Hall and Wilson, 1991] corrections in
which the bootstrap estimate is Studentized. For the one-sample case, we
want an interval estimate based on the distribution of (
b
- )/
s
b
, where
and
b
are the estimates of the unknown parameter based on the original
and bootstrap sample, respectively, and
s
b
denotes the standard deviation of
the bootstrap sample. An estimate of the population variance is required
to transform the resultant interval into one about q(see Carpenter and
Bithell [2000]).
For the two-sample case, we want a confidence interval based on the
distribution of
q
q
q
q
s
(
)
ˆ
ˆ
qq
nb
-
mb
,
(
)
+-
(
)
2
2
ns ms
nm
-
1
1
(
)
nb
mb
11
nm
+
+-
2
where
n
,
m
, and
s
nb
,
s
mb
denote the sample sizes and standard deviations,
respectively, of the bootstrap samples. Applying the Hall-Wilson correc-
tions, we obtain narrower interval estimates that are more likely to contain
the true value of the unknown parameter.
The bias-corrected and accelerated BC
a
interval due to Efron and
Tibshirani [1986] also represents a substantial improvement, though for
