Resampling approaches in multi-sample problems

Question

I'm new to bootstrap and I would need an explanation about resampling techniques for comparing two populations. Suppose we have two samples, of respective sample sizes $n_1$ and $n_2$. The total number of individuals is $n = n_1 + n_2$.

We want to use a bootstrap/resampling technique to compare a given statistic (say, the coefficient of variation) between these two samples.

There is the standard approach: drawing with replacement $B$ bootstrap samples of respective sample sizes $n_1$ and $n_2$, calculating the statistic in each bootstrap sample, etc.

There could be another approach: drawing (still with replacement) directly $B$ bootstrap samples of $n$ individuals from the total sample, regardless of the groups. And then, calculating $B$ times the statistic in each sub-group, etc.

My intuition is that the second approach is flawed, but I couldn't explain why in a formal manner. Is such an approach acceptable in some cases / for some statistics, or is it always wrong, and why?

Thanks!

Answer 1

Unless the two populations have the same characteristics it would be incorrect to take the B bootstrap samples regardless of the group. As an example, if your sample set was the average wage of a 25 year old professional, this data could vary significantly depending upon the industry and the geography the professional is in.

Given the divergence it wouldn't make much sense to group all the samples into a single population. For the same reason it wouldn't make sense to collect the bootstrap samples across all the different population.