Bootstrap variability: $n$-out-of-$n$ vs. $m$-out of-$n$

Question

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 $Y_{(1)} - X_{(n)}$.

Answer 1

I think that you are reading a bit too much into an heuristic introduction to why the m-out-of-n bootstrap might make sense.*

I don't see anything "counterintuitive" as I read the first quotation. What I see is a simple re-statement of something like what you know about the standard error of the mean, were you to estimate the mean by bootstrapping with $m \ne n$. At a constant finite population variance, the standard error of the mean will be smaller based on a larger number of observations, and larger if based on a smaller number of observations. The classical central limit theorem at work here is based on an assumption of independent and identically distributed (IID) observations, but there are extensions to some types of dependences that have similar implications for the relationship between sample size and variability of estimates.

The second quote explicitly looks at what might be found with particular types of dependencies. That's an interesting and instructive contrast from the (implied) IID situation of the first quote. The second quote doesn't seem, however, to apply directly to your use case, which I understand to have IID $(X,Y)$ pairs as observations. Quoting from farther down in that section of the book:

an m-out-of-n bootstrap with m << n ... also works in independent cases ... when estimating the asymptotic behavior of the maximum of an independent and identically distributed (i.i.d.) sequence of random variables.

That seems to fit your use case. An important consideration is finding an appropriate value of m in that situation. If your colleagues followed, for example, Bickel and Sakov, "On The choice of m in the m out of n bootstrap and confidence bounds for extrema," Statistica Sinica 18(2008), 967-985, I don't see that a reviewer should question your statistical rationale.

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*Should Michael Chernick come across this, perhaps he could weigh in on what he and his co-author meant. Quotes in the question and this answer can be found in: Michael R. Chernick and Robert A. LaBudde, "An Introduction to Bootstrap Methods with Applications to R," Wiley, 2011.