Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-14 CHAPTER 14 Bootstrap Methods and Permutation Tests
bias
bootstrap
estimate of bias
• Center: A statistic is biased as an estimate of the parameter if its sampling
distribution is not centered at the true value of the parameter. We
can check bias by seeing whether the bootstrap distribution of the statistic
is centered at the value of the statistic for the original sample.
More precisely, the bias of a statistic is the difference between the mean
of its sampling distribution and the true value of the parameter. The bootstrap
estimate of bias is the difference between the mean of the bootstrap
distribution and the value of the statistic in the original sample.
• Spread: The bootstrap standard error of a statistic is the standard deviation
of its bootstrap distribution. The bootstrap standard error estimates
the standard deviation of the sampling distribution of the statistic.
Bootstrap t confidence intervals
If the bootstrap distribution of a statistic shows a normal shape and small
bias, we can get a confidence interval for the parameter by using the bootstrap
standard error and the familiar t distribution. An example will show how
this works.
We are interested in the selling prices of residential real estate in
EXAMPLE 14.4
Seattle, Washington. Table 14.1 displays the selling prices of a random
sample of 50 pieces of real estate sold in Seattle during 2002, as recorded by the
county assessor. 6 Unfortunately, the data do not distinguish residential property from
commercial property. Most sales are residential, but a few large commercial sales in
a sample can greatly increase the sample mean selling price.
Figure 14.6 shows the distribution of the sample prices. The distribution is far
from normal, with a few high outliers that may be commercial sales. The sample is
small, and the distribution is highly skewed and “contaminated” by an unknown number
of commercial sales. How can we estimate the center of the distribution despite
these difficulties?
The first step is to abandon the mean as a measure of center in favor of a
statistic that is more resistant to outliers. We might choose the median, but
in this case we will use a new statistic, the 25% trimmed mean.
TABLE 14.1
Sellingprices for Seattle real estate, 2002 ($1000s)
142 175 197.5 149.4 705 232 50 146.5 155 1850
132.5 215 116.7 244.9 290 200 260 449.9 66.407 164.95
362 307 266 166 375 244.95 210.95 265 296 335
335 1370 256 148.5 987.5 324.5 215.5 684.5 270 330
222 179.8 257 252.95 149.95 225 217 570 507 190