Determining sample size necessary for bootstrap method / Proposed Method

I know this is a rather hot topic where no one really can give a simple answer for. Nevertheless I am wondering if the following approach couldn’t be useful.

The bootstrap method is only useful if your sample follows more or less (read exactly) the same distribution as the original population. In order to be certain this is the case you need to make your sample size large enough. But what is large enough?

If my premise is correct you have the same problem when using the central limit theorem to determine the population mean. Only when your sample size is large enough you can be certain that the population of your sample means is normally distributed (around the population mean). In other words, your samples need to represent your population (distribution) well enough. But again, what is large enough?

In my case (administrative processes: time needed to finish a demand vs amount of demands) I have a population with a multi-modal distribution (all the demands that are finished in 2011) of which I am 99% certain that it is even less normally distributed than the population (all the demands that are finished between present day and a day in the past, ideally this timespan is as small as possible) I want to research.

My 2011 population exists out of enough units to make $$x$$ samples of a sample size $$n$$. I choose a value of $$x$$, suppose $$10$$ ($$x=10$$). Now I use trial and error to determine a good sample size. I take an $$n=50$$, and see if my sample mean population is normally distributed by using Kolmogorov-Smirnov. If so I repeat the same steps but with a sample size of $$40$$, if not repeat with a sample size of $$60$$ (etc.).

After a while I conclude that $$n=45$$ is the absolute minimum sample size to get a more or less good representation of my 2011 population. Since I know my population of interest (all the demands that are finished between present day and a day in the past) has less variance I can safely use a sample size of $$n=45$$ to bootstrap. (Indirectly, the $$n=45$$ determines the size of my timespan: time needed to finish $$45$$ demands.)

This is, in short, my idea. But since I am not a statistician but an engineer whose statistics lessons took place in the days of yonder I cannot exclude the possibility I just generated a lot of rubbish :-). What do you guys think? If my premise makes sense, do I need to chose an $$x$$ larger than $$10$$, or smaller? Depending on your answers (do I need to feel embarrassed or not? :-) I'll be posting some more discussion ideas.

But I am afraid that in my attempt to give information I completely clouded my question. I know that the bootstrap samples take over the distribution of the population sample. I follow you completely but...

Your original population sample needs to be large enough to be moderately certain that the distribution of your population sample corresponds (equals) with the 'real' distribution of the population.

This is merely an idea on how to determine how large your original sample size needs to be in order to be reasonably certain that the sample distribution corresponds with the population distribution.

Suppose you have a bimodal population distribution and one top is a lot larger than the other one. If your sample size is 5 the chance is large that all 5 units have a value very close to the large top (chance to ad randomly draw a unit there is the largest). In this case your sample distribution will look unimodal.

With a sample size of a hundred the chance that your sample distribution is also bimodal is a lot larger!! The trouble with bootstrapping is that you only have one sample (and you build further on that sample). If the sample distribution really does not correspond with the population distribution you are in trouble. This is just an idea to make the chance of having 'a bad sample distribution' as low as possible without having to make your sample size infinitely large.

• checkout bayesian bootstrap sampling which might cope with small sample size. See sumsar.net/blog/2015/04/… for more details. Apr 29 '19 at 9:15

I took interest in this question because I saw the word bootstrap and I have written books on the bootstrap. Also people often ask "How many bootstrap samples do I need to get a good Monte Carlo approximation to the bootstrap result?" My suggested answer to that question is to keep increasing the size until you get convergence. No one number fits all problems.

But that is apparently not that question you are asking. You seem to be asking what the original sample size needs to be for the bootstrap to work. First of all I do not agree with your premise. The basic nonparametric bootstrap assumes that the sample is taken at random from a population. So for any sample size $n$ the distribution for samples chosen at random is the sampling distribution assumed in bootstrapping. The bootstrap principle says that choosing a random sample of size $n$ from the population can be mimicked by choosing a bootstrap sample of size $n$ from the original sample. Whether or not the bootstrap principle holds does not depend on any individual sample "looking representative of the population". What it does depend on is what you are estimating and some properties of the population distribution (e.g., this works for sampling means with population distributions that have finite variances, but not when they have infinite variances). It will not work for estimating extremes regardless of the population distribution.

The theory of the bootstrap involves showing consistency of the estimate. So it can be shown in theory that it works for large samples. But it can also work in small samples. I have seen it work for classification error rate estimation particularly well in small sample sizes such as 20 for bivariate data.

Now if the sample size is very small---say 4---the bootstrap may not work just because the set of possible bootstrap samples is not rich enough. In my book or Peter Hall's book this issue of too small a sample size is discussed. But this number of distinct bootstrap samples gets large very quickly. So this is not an issue even for sample sizes as small as 8. You can take a look at these references:

• Is there any standard test to run to check if there (for example 4 samples) is not enough? I have a dataset where I'm calculating bootstrapped confidence intervals for the mean, but some individuals have very few datapoints (< 8 in some cases). My instinct tells me that I should disregard individuals who have fewer than n data points, but then how do I define this cuttoff n? I was hoping to find a generally accepted cutoff value (analogous to how 6 or 7 is the arbitrary cut-off point for the number of samples per group in a mixed-model analysis). Apr 28 '17 at 13:03

The resampling process creates many possible samples that a study could have drawn. The various combinations of values in the simulated samples collectively provide an estimate of the variability between random samples drawn from the same population. The range of these potential samples allows the procedure to construct confidence intervals and perform hypothesis testing.

Importantly, as the bootstrap sample size increases, bootstrapping converges on the correct sampling distribution under most conditions.

In regards to your question on: "This is merely an idea on how to determine how large your original sample size needs to be in order to be reasonably certain that the sample distribution corresponds with the population distribution."

This is dependent on the specific problem that you are examining and is not dependent on the bootstrap sample size. The purpose of the bootstrap sample is merely to obtain a large enough bootstrap sample size, usually at least 1000 in order to obtain with low MC errors such that one can obtain distribution statistics on the original sample e.g. 95% CI. But this cannot guarantee that the original sample taken is representative of the actual population distribution.