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Consider two samples ${(x_i)}_{i=1}^m$ and ${(y_i)}_{i=1}^n$. Assume that the $x_i$ are independent replicates from a distribution with expectation $\mu_X$ and similarly the $y_i$ are independent replicates from a distribution with expectation $\mu_Y$, and also assume that the two samples are independent. Is it possible to get a boostrap confidence interval about the difference of means $\mu_X-\mu_Y$ ? Or is there another nonparametric way to get such a confidence interval ?

EDIT : Oops - I've just seen this topic Confidence interval for the difference of two means using boot package in R Nevertheless I am interested in understanding why the method is correct. This is not a classical bootstrap procedure, isn't it ? Here we sample separately in each data sample. This is not like the classical bootstrap which is the case when there is only one data sample.

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Yes. It is like a stratified bootstrap. You sample with replacement m times for population 1 and calculate a bootstrap mean for it. Do the same by sampling with replacement n times from population 2 calculate a bootstrap mean for it and then take the difference of the bootstrap means. Repeat this many time to get an approximate bootstrap distribution for the mean difference. As long as the conditions for the bootstrap mean to work are satisfied for each population the bootstrap will work on the mean difference. In my book Bootstrap Methods 2nd Edition pp. 67-71, I show an example where I did this in a clinical trial looking at the mean difference in capture thresholds for two pacing leads.

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