Some Notes Pertaining to Ch. 5 of E&T
Sec. 5.1 suggests that while summary statistics are often obtained near the
start of a data analysis, some measure of their accuracy may typically follow.
(I typically prefer obtaining a confidence interval instead of being content with a
crude measure of accuracy such as an estimated standard error, since the standard error doesn't
account for the role of bias in inaccuracy, and also an estimated standard error may of course
itself be inaccurate.)
This chapter considers the simple situation of estimating the standard error of the sample mean,
which is a situation where bootstrapping isn't needed since it (almost) reduces to the usual estimate
of the standard error (of the sample mean).
Note the somewhat unusual notation introduced by (5.2) on p. 39.
Hopefully you are very familiar with the results given by (5.3), (5.4), (5.5), and (5.6) in Sec. 5.2.
(For large sample sizes, the results stated in (5.6) lead to approximate confidence intervals for the
distribution mean if we replace the true distribution standard deviation by an estimate (assuming the actual
standard deviation is unknown).)
The plug-in estimate of the distribution standard deviation given by (5.11) and the standard error expression given
by (5.4) combine to yield the estimated standard error given by (5.12), which is just the
plug-in estimate of the standard error of the sample mean. Note that it is
slightly different from the usual standard error formula.
(Note: For some distributions, some other way of estimating the standard error of the sample mean way be better,
but if we don't have much knowledge about the specific nature of the underlying distribution it's dangerous to
replace the usual estimate of the distribution standard deviation by an alternative, since the alternative may be
rather biased.)
In this chapter we can see that the plug-in estimate of the standard error of the sample mean leads to nearly the same thing
we get from a traditional approach. (For those of you familiar with the material of STAT 652, it can be
noted that the plug-in estimate is exactly a method of moments estimate.) What prevents us from using a plug-in
estimate of the standard error for other estimators is the fact that we typically don't have a formula like (5.4)
for other estimators. However, it turns out that using bootstrapping to obtain an estimated standard error for some
specific estimator has a connection to the plug-in principle.