Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-16 CHAPTER 14 Bootstrap Methods and Permutation Tests
TRIMMED MEAN
A trimmed mean is the mean of only the center observations in a data
set. In particular, the 25% trimmed mean x 25% ignores the smallest
25% and the largest 25% of the observations. It is the mean of the
middle 50% of the observations.
Recall that the median is the mean of the 1 or 2 middle observations. The
trimmed mean often does a better job of representing the average of typical
observations than does the median. Our parameter is the 25% trimmed mean
of the population of all real estate sales prices in Seattle in 2002. By the plug-in
principle, the statistic that estimates this parameter is the 25% trimmed mean
of the sample prices in Table 14.1. Because 25% of 50 is 12.5, we drop the 12
lowest and 12 highest prices in Table 14.1 and find the mean of the remaining
26 prices. The statistic is (in thousands of dollars)
x 25% = 244.0019
We can say little about the sampling distribution of the trimmed mean
when we have only 50 observations from a strongly skewed distribution. Fortunately,
we don’t need any distribution facts to use the bootstrap. We bootstrap
the 25% trimmed mean just as we bootstrapped the sample mean: draw
1000 resamples of size 50 from the 50 selling prices in Table 14.1, calculate
the 25% trimmed mean for each resample, and form the bootstrap distribution
from these 1000 values.
Figure 14.7 shows the bootstrap distribution of the 25% trimmed mean.
Here is the summary output from S-PLUS:
Number of Replications: 1000
Summary Statistics:
Observed Mean Bias SE
TrimMean 244 244.7 0.7171 16.83
What do we see? Shape: The bootstrap distribution is roughly normal. This
suggests that the sampling distribution of the trimmed mean is also roughly
normal. Center: The bootstrap estimate of bias is 0.7171, small relative to the
value 244 of the statistic. So the statistic (the trimmed mean of the sample)
has small bias as an estimate of the parameter (the trimmed mean of the population).
Spread: The bootstrap standard error of the statistic is
SE boot = 16.83
This is an estimate of the standard deviation of the sampling distribution of
the trimmed mean.
Recall the familiar one-sample t confidence interval (page 452) for the
mean of a normal population:
x ± t ∗ SE = x ± t ∗ √ s
n
This interval is based on the normal sampling distribution of the sample mean
x and the formula SE = s/ √ n for the standard error of x. When a bootstrap