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?