In statistics, there are existing formulas for the sample size with confidence intervals to estimate parameters (mean, proportion…) of the population (see the link). But if we add bootstrapping into such process, is there any exact formula? I notice there are books or discussions saying that the sample size cannot be too small. But I just wonder is there any direct formula to calculate the sample size?
https://www.isixsigma.com/tools-templates/sampling-data/how-determine-sample-size-determining-sample-size/
This may be useful
Computer-Intensive Statistics c 2009–11 B. D. Ripley
https://warwick.ac.uk/fac/sci/statistics/apts/students/resources-2010-2011/compintens_notes.pdf
see section on How many bootstrap resamples - crucially it will depend on what you are trying to estimate eg need more for standard error than a mean- dealing with tails not the middle; pragmatic do it in batches of say 250 and plot the cumulative average to convergence; computing and time is relatively cheap ; thinking and data collection isn’t.
This may be useful
Computer-Intensive Statistics c 2009–11 B. D. Ripley
https://warwick.ac.uk/fac/sci/statistics/apts/students/resources-2010-2011/compintens_notes.pdf
see section on How many bootstrap resamples - crucially it will depend on what you are trying to estimate eg need more for standard error than a mean- dealing with tails not the middle; pragmatic do it in batches of say 250 and plot the cumulative average to convergence; computing and time is relatively cheap ; thinking and data collection isn’t.
The reply by Kelvyn is good and should suit nearly all practical purposes. But you should recall that bootstrapping is essentially random sampling from a finitely-sized set of possible bootstrap samples. If the original data sample is small, the number of possible bootstrap samples, while much larger might still of a size where the difference between sampling-with-replacement and sampling-without-replacement becomes important. Most bootstrapping is, I think, done by sampling-with-replacement, as this is simplest to implement. But if you are starting from scratch, and have a very small-sized original sample, it may be worth dealing with with separately. For example, if your original sample size is only 5 o 6, the number of possible bootstrap samples only 3125 or 46656 and these are with range, with modern computers, of doing computations with the complete set of these.
The reference provides formulae for the variances (due to resamping) of estimates obtained by resampling with and without replacemant,in case thesse are useful to you: http://home.iitk.ac.in/~shalab/sampling/chapter2-sampling-simple-random-sampling.pdf
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