Now that we have some idea of how (Efron style) bootstrap confidence intervals
work, we can return to the subsampling bootstrap and look at the kind of
confidence intervals Politis and Romano (1994) recommend for the subsampling
bootstrap.
Somewhat suprisingly, they recommend the kind of intervals that Efron and
Tibshirani disparage in Section 13.4 titles Is the Percentile Interval
Backwards?. This section is a response to the recommendations of Hall
(The Bootstrap and Edgeworth Expansion, Springer, 1992, and earlier
papers) where he charactizes bootstrap percentile intervals as looking up
[in] the wrong statistical tables backwards.
We take no sides on the argument between Efron and Hall. There are arguments
on both sides, and either method performs well in some examples and badly in
others.
However, for the subsampling bootstrap, there seems to be no dispute.
The standard method for the subsampling bootstrap is very much like
what Hall recommends for the ordinary bootstrap.
This method is explained in a handout (which will be handed out in class).
If you want to see what it looks like in various formats
Everything down to the bottom of the for loop should be familiar,
just like what we did when we previously
visited this example except some plot commands have been removed.
The four lines below the for calculate the subsampling
bootstrap confidence interval. The example could have stopped there.
The histogram shows the distribution which the subsampling bootstrap uses
to calculate critical values. And the confidence interval below gives
what one could do if one wanted to assume the asymptotic distribution is
normal.
Note that the histogram is quite skewed. Hence we get a confidence interval
that is not symmetric about theta.hat. Thus we do much better
by not assuming normality. We get a shorter confidence interval with better
coverage.
As usual, Efron and Tibshirani are using a ridiculously small sample size
in this toy problem. There is no reason to believe the subsampling bootstrap
here. But it is reasonable for (much) larger data sets.
(This comment repeated from the previous discussion of the subsampling
bootstrap).
Everything down to the bottom of the for loop should be familiar,
just like what we did when we previously
visited this example except some the commands to also do the Efron
bootstrap have been removed.
The code immediately following the for loop calculates
a two-sided confidence interval as described in the handout and in
Politis and Romano (1994). That is what you would usually do.
In this problem we actually want a one-sided interval because we know
the estimator always misses the parameter on the low side. That's done
by the lines below the comment.