I am curious about bootstrap methods, and just read the very elementary explanation of them here:
http://bcs.whfreeman.com/ips5e/conte...df/moore14.pdf
It is very interesting to read, but leaves me with a lot of questions. For instance, the
claim is that the bootstrap distribution approximates the shape and spread of the full sampling distribution. But this is obviously not true in complete generality--you can certainly choose a pathological sample from a population that will yield a bootstrap
that doesn't approximate either one at all well. So how can this be quantified?
Are there any theorems along the lines of the following? "Given an underlying distribution satisfying properties A, B, C, and a statistic satisfying properties D, E, and F, the probability of obtaining a sample of size n for which the sampling distribution is within some distance of the bootstrap distribution (appropriately shifted) in some appropriate sense."
or,
"given an underlying distribution satisfying properties A,B, C, and
a statistic satisfying D, E and F, then as n->infinity,
the bootstrap distribution for (all but a set of samples whose measure shrinks to 0) approaches the sampling distribution of size n in some appropriate sense."
Alternatively, are there simulations that suggest the bootstrap works well in various situations? When doesn't it work well (other than for small samples, of course)?
Are there particular statistics or underlying distributions for which it is known to
go way off?
Is there a good reference for this that isn't 200 pages long and containing 256923452365432345 different variable names (which is what most of the books in my area of maths are like--but I am doing this for recreation!)?
http://bcs.whfreeman.com/ips5e/conte...df/moore14.pdf
It is very interesting to read, but leaves me with a lot of questions. For instance, the
claim is that the bootstrap distribution approximates the shape and spread of the full sampling distribution. But this is obviously not true in complete generality--you can certainly choose a pathological sample from a population that will yield a bootstrap
that doesn't approximate either one at all well. So how can this be quantified?
Are there any theorems along the lines of the following? "Given an underlying distribution satisfying properties A, B, C, and a statistic satisfying properties D, E, and F, the probability of obtaining a sample of size n for which the sampling distribution is within some distance of the bootstrap distribution (appropriately shifted) in some appropriate sense."
or,
"given an underlying distribution satisfying properties A,B, C, and
a statistic satisfying D, E and F, then as n->infinity,
the bootstrap distribution for (all but a set of samples whose measure shrinks to 0) approaches the sampling distribution of size n in some appropriate sense."
Alternatively, are there simulations that suggest the bootstrap works well in various situations? When doesn't it work well (other than for small samples, of course)?
Are there particular statistics or underlying distributions for which it is known to
go way off?
Is there a good reference for this that isn't 200 pages long and containing 256923452365432345 different variable names (which is what most of the books in my area of maths are like--but I am doing this for recreation!)?