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only when the location parameters of the two distributions are known or
are known to be equal.
The test proposed by Miller [1968] yields conservative Type I errors,
less than or equal to the declared error, unless the sample sizes are
unequal. A 10% test with samples of size 12 and 8 taken from normal
populations yielded Type I errors 14% of the time.
Fligner and Killeen [1976] propose a permutation test based on the
sum of the absolute deviations from the combined sample mean. Their
test may be appropriate when the medians of the two populations are
equal, but can be virtually worthless otherwise, accepting the null hypoth-
esis up to 100% of the time. In the first edition, Good [2001] proposed a
test based on permutations of the absolute deviations from the individual
sample medians; this test, alas, is only asymptotically exact and even then
only for approximately equal sample sizes, as shown by Baker [1995].
To compute the primitive bootstrap introduced by Efron [1979], we
would take successive pairs of samples—one of
n
observations from the
sampling distribution
F
n
which assigns mass 1/
n
to the values {
X
i
:
i
= 1,
...,
n
}, and one of
m
observations from the sampling distribution
G
m
which assigns mass 1/
m
to the values {
X
j
:
j
=
n
+ 1,...,
n
+
m
}, and
compute the ratio of the sample variances
(
)
s
2
n
-
1
1
n
R
=
.
(
)
s
2
m
-
m
We would use the resultant bootstrap distribution to test the hypothesis
that the variance of
F
equals the variance of
G
against the alternative that
the variance of
G
is larger. Under this test, we reject the null hypothesis if
the 100(1 - a) percentile is less than 1.
This primitive bootstrap and the associated confidence intervals are close
o exact only for very large samples with hundreds of observations. More
often the true coverage probability is larger than the desired value.
Two corrections yield vastly improved results. First, for unequal-sized
samples, Efron [1982] suggests that more accurate confidence intervals
can be obtained using the test statistic
2
sn
sm
n
R
¢=
2
m
Second, applying the bias and acceleration corrections described in
Chapter 3 to the bootstrap distribution of
R
¢ yields almost exact
intervals.
