Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-42 CHAPTER 14 Bootstrap Methods and Permutation Tests
While the BCa and tilting calculations are radically different, the results
tend to be about the same, except for random variation in the BCa if the number
of resamples is less than about 5000. Both procedures are accurate, so we
expect them to produce similar results unless a small sample size makes any
inference dubious.
SECTION 14.4
Summary
Both bootstrap t and (when they exist) traditional z and t confidence intervals
require statistics with small bias and sampling distributions close to normal.
We can check these conditions by examining the bootstrap distribution for
bias and lack of normality.
The bootstrap percentile confidence interval for 95% confidence is the interval
from the 2.5% percentile to the 97.5% percentile of the bootstrap distribution.
Agreement between the bootstrap t and percentile intervals is an
added check on the conditions needed by the t interval. Do not use t or percentile
intervals if these conditions are not met.
When bias or skewness is present in the bootstrap distribution, use either a
BCa or bootstrap tilting interval. The t and percentile intervals are inaccurate
under these circumstances unless the sample sizes are very large. The
tilting and BCa confidence intervals adjust for bias and skewness and are
generally accurate except for small samples.
SECTION 14.4
Exercises
Many of these exercises require software that will calculate accurate bootstrap
confidence intervals. If your software finds BCa but not tilting intervals, ignore
requests for tilting intervals. S-PLUS supplies both types.
14.26 What percentiles of the bootstrap distribution are the endpoints of a 90%
bootstrap percentile confidence interval?
14.27 In Exercise 14.17 (page 14-25) you bootstrapped the mean of a simulated SRS
from the standard normal distribution N(0, 1) and found the standard t and
bootstrap t 95% confidence intervals for the mean.
(a) Find the bootstrap percentile 95% confidence interval. Does this interval
confirm that the t intervals are acceptable?
(b) We know that the population mean is 0. Do the confidence intervals capture
this mean?
14.28 Bootstrapping is a good way to check if traditional inference methods are accurate
for a given sample. Consider the following data:
109 123 118 99 121 134 126 114 129 123 171 124
111 125 128 154 121 123 118 106 108 112 103 125
137 121 102 135 109 115 125 132 134 126 116 105
133 111 112 118 117 105 107