# Bootstrapping a sample from a finite population

Can someone point me to some reference for theory on bootstrapping a sample took from a population of known size?

I am used to use Bootstrap to calculate confidence intervals of a sample when the population size is considered way larger than the sample (therefore a random selection with repetition should emulate well the sampling process).

Now say I know the population is 1000, and I sampled 800 (and let's assume the sampling is in fact random). Random selection with repetition does not seem to be appropriate. By pigeonhole principle, if I truly take another random sample of size 800, it is guaranteed that at least 600 values will be the same as the original sample, something traditional bootstrap cannot replicate (and might miss by a lot).

Any solutions? I thought of:

• Sampling 1000 with repetition, then randomly picking 800 (seems to be an equivalent approach of traditional bootstrap)
• Sample 600 without repetition, than sampling 200 more using all 800 samples with repetition. This would account for the effect I described earlier.

Any thoughts on what is good and bad with those approaches? Or any alternative approach?

Bootstrap sampling should resemble the process of sampling the data from the population. In case of finite population you sampled fraction $f$ out of population of size $N$, i.e. $n = fN$ cases. There are two problems with using bootstrap in such scenario: (1) if you used traditional bootstrap, you'd be sampling with replacement rather than without replacement, (2) if you sampled without replacement $fn$ cases, then you'd end up with sample smaller than $n$. The first scenario is a bad idea since in such case bootstrap would not resemble the original sampling process. For using bootstrap in finite population case you have three alternatives:

1. Sample without replacement samples of size $fn$ and then rescale the results. Finding the appropriate rescaling factor can be more complicated then it sounds, so this may not be the best alternative.
2. First sample without replacement $N-n$ cases out of your sample, concatenate them to the sample, and then sample without replacement $n$ cases out of it. This is called mirror-match bootstrap.
3. First sample with replacement $N$ cases out of your sample, and then sample out of it $n$ cases without replacement. This is called superpopulation bootstrap.