Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-34 CHAPTER 14 Bootstrap Methods and Permutation Tests
(b) Draw an SRS of size n = 10 from this population. Bootstrap the sample
mean x using 1000 resamples from your sample. Give a histogram of the
bootstrap distribution and the bootstrap standard error.
(c) Repeat the same process for samples of sizes n = 40 and n = 160.
(d) Write a careful description comparing the three bootstrap distributions.
What are the effects of increasing the sample size?
14.25 The populations in the two previous exercises have the same mean and standard
deviation, but one is very close to normal and the other is strongly nonnormal.
Based on your work in these exercises, how does nonnormality of the
population affect the bootstrap distribution of x? How does it affect the bootstrap
standard error? Does either of these effects diminish when we start with
a larger sample? Explain what you have observed based on what you know
about the sampling distribution of x and the way in which bootstrap distributions
mimic the sampling distribution.
14.4 Bootstrap Confidence Intervals
To this point, we have met just one type of inference procedure based on resampling,
the bootstrap t confidence intervals. We can calculate a bootstrap
t confidence interval for any parameter by bootstrapping the corresponding
statistic. We don’t need conditions on the population or special knowledge
about the sampling distribution of the statistic. The flexible and almost automatic
nature of bootstrap t intervals is appealing—but there is a catch. These
intervals work well only when the bootstrap distribution tells us that the sampling
distribution is approximately normal and has small bias. How well must
these conditions be met? What can we do if we don’t trust the bootstrap t interval?
In this section we will see how to quickly check t confidence intervals
for accuracy and learn alternative bootstrap confidence intervals that can be
used more generally than the bootstrap t.
Bootstrap percentile confidence intervals
Confidence intervals are based on the sampling distribution of a statistic. If a
statistic has no bias as an estimator of a parameter, its sampling distribution
is centered at the true value of the parameter. We can then get a 95% confidence
interval by marking off the central 95% of the sampling distribution.
The t critical values in a t confidence interval are a shortcut to marking off the
central 95%. The shortcut doesn’t work under all conditions—it depends both
on lack of bias and on normality. One way to check whether t intervals (using
either bootstrap or formula-based standard errors) are reasonable is to compare
them with the central 95% of the bootstrap distribution. The 2.5% and
97.5% percentiles mark off the central 95%. The interval between the 2.5%
and 97.5% percentiles of the bootstrap distribution is often used as a confidence
interval in its own right. It is known as a bootstrap percentile confidence
interval.