I asked a recent question on CV regarding failure of the bootstrap using the sample range as an illustration. That question was not very clear, and so the question was closed.
I am now asking a new, but related, simpler question.
After some reading, the $m$-out-of-$n$ bootstrap stands out as a potential solution to the traditional ($n$-out-of-$n$) bootstrap by Chernick (2007, 2011).
Basically, instead of taking resamples of the same size as the original data, a smaller number ($m$) of resamples is drawn. Simple enough and this method has been demonstrated to work for both IID and dependent situations.
Chernick (2007, 2011) chapter 6.5 says in part:
"... choosing a sample size $m$ larger than $n$ would lead to an estimate with less variability than the original sample, while a sample $m$ smaller than $n$ would cause the estimator to have more variability."
The above quote sounds counterintuitive and would raise some eyebrows from those less familiar with statistical resampling methods.
"... sampling with replacement in the dependent situations creates estimates with variability less than what we actually have in the original data set since $n$-dependent observation contain only the information equivalent to a smaller number of independent observations."
My question(s): Does the first quotation above refer to the IID case? Thus the variability of the $m$-out-of-$n$ bootstrap in comparison to the $n$-out-of-$n$ depends strongly on whether the original data are independent or not?
This seems like the case, since the bootstrap presumes samples are IID in the first place, but I would like to be sure. The reason I ask is because my colleagues and I are writing up a manuscript and we want to give some rationale for choosing the $m$-out-of-$n$ bootstrap.
Context: For my particular situation, I am needing to estimate the sampling distribution of $Y_{(1)} - X_{(n)}$, where both ($X$, $Y$) $\geq 0$ and ($X$, $Y$) $\in$ (0, 1). That is the minimum of $Y$ minus the maximum of $X$. This statistic is a function of extreme order statistics, which are known to be problematic for the usual bootstrap procedure. I see this quantity as akin to the sample range (except calculated using two distinct datasets), hence my previous closed question.
Further, I know nothing of the joint probability distribution of $Y_{(1)} - X_{(n)}$.