Applied Statistics Using SPSS, STATISTICA, MATLAB and R

3.6 Bootstrap Estimation 101
about the bootstrap technique is that it also often works for other statistics for
which no theory on sampling distribution is available. As a matter of fact, the
bootstrap distribution usually – for a not too small original sample size, say n > 50
− has the same shape and spread as the original sampling distribution, but is
centred at the original statistic value rather than the true parameter value.
12
n
10
8
6
4
2
300
n
x
x *
0
0
a 100 200 300 400 500 600 700b
300 320 340 360 380 400 420
Figure 3.7. a) Histogram of the PRT data; b) Histogram of the bootstrap means.
Suppose that the bootstrap distribution of a statistic, w, is approximately normal
and that the bootstrap estimate of bias is small. We then compute a two-sided
bootstrap confidence interval at α risk, for the parameter that corresponds to the
statistic, by the following formula:
w ± tn
−1,
1−α
/ 2
SE
boot
We may use the percentiles of the normal distribution, instead of the Student’s t
distribution, whenever m is very large.
The question naturally arises on how large must the number of bootstrap
samples be in order to obtain a reliable bootstrap distribution with reliable values
of SEboot? A good rule of thumb for m, based on theoretical and practical evidence,
is to choose m ≥ 200.
The following examples illustrate the computation of confidence intervals using
the bootstrap technique.
Example 3.9
Q: Consider the percentage of lime, CaO, in the composition of clays, a sample of
which constitutes the Clays’ dataset. Compute the confidence interval at 95%
level of the two-tail 5% trimmed mean and discuss the results. (The two-tail 5%
trimmed mean disregards 10% of the cases, 5% at each of the tails.)
250
200
150
100
50