# Might the Nyquist theorem point toward an answer for the number of bootstrap samples required?

It seems like the generally accepted answer to, "How many bootstrap replications should I run?" has been, "It depends." It depends seems like a correct answer. Nevertheless, I was wondering whether there might be a parallel between bootstrap observations and samples taken from a continuous process in such a way as to allow for a better answer.

Specifically, the Nyquist theorem appears to suggest one sample at at least twice the occupied bandwidth. Density plots of observed (or bootstrapped) distributions also have 'bandwidth'. Can these two types of bandwidth be equated for the purposes of specifying a desired degree of accuracy? That is, if attempting to get a bootstrapped distribution that is sufficient (maybe excessive) could one terminate once one had no fewer than two observations within each (perhaps rolling?) bin that is the desired degree of accuracy?

For point estimates it would then seem to be fully sufficient (maybe excessive) to sample until one collects two bootstrap samples within a minimum bin size of interest, e.g. if one wanted a score that was accurate to the hundredths place, then the estimate and its two nearest-sampled neighbors should be at least within .005 of each other – of course that assumes that you know where your point estimate will be, which you never will. So then, if you had a prior point estimate, you could wait until that condition was satisfied and then sample twice again as much to be 'sure'. Alternatively, if you really care about the fidelity of the entire distribution, then you could sample until no two observations are further apart than your nearest-neighbor threshold (eliminating observations that are identical). Of course it would be important to consider that some thresholds can never be met (e.g. a bootstrap of the mean of 1, 2, and 3 below ⅓‎).

This (of course) doesn't cover all types of bootstraps that one might want to perform, but it seems like it might cover some. This might end up being a matter of opinion, but I'm hoping there is an answer or informed opinion sufficient to keep the question alive.

• For something related, but not quite germane, you might look at Bernard Widrow's 1956 ScD thesis, A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory (MIT). There is a separate journal article with the same author and title. At the risk of sounding somewhat unappreciative, it is essentially just an application of Nyquist theory along the amplitude dimension as opposed to the time dimension. May 8, 2014 at 2:39
• May 8, 2014 at 2:44
• This question is a misguided. And the other answers given here don't address the misconceptions in this question at all. The bootstrap is about "sampling" in the statistical sense i.e how to estimate uncertainty in estimates of your model given a random sample of observations. On the other hand, Nyquist sampling does not require any notion of randomness. It is all about discretizing a continuous periodic signal that could be entirely deterministic. Apr 29, 2015 at 21:38