Bootstrapped difference test with smaller subsamples Question
How do I perform a bootstrapped difference test when my bootstrapped sample sizes are smaller than the original sample sizes?
Typically I would do this by taking  $p = \frac{\sum_{i=1}^{B} t_{i}^ {*} \geq t}{B}$, where

*

*$t_{i}^ {*}=\frac{\bar{x^ {*} }-\bar{y}^ {*}}{\sqrt{\sigma^{*2}_x/m + \sigma^{*2}_y/m}}$ for a pair of bootstrapped samples $x^{*}$, $y^{*}$

*$t=\frac{\bar{x}-\bar{y}}{\sqrt{\sigma^{2}_y/m + \sigma^{2}_x/m}}$
In this case, however, the obvious substitute for $t$ is

*

*$t=\frac{\bar{x}-\bar{y}}{\sqrt{\sigma^{2}_x/n_1 + \sigma^{2}_y/n_2}}$, where $m \neq n_1 \neq n_2$
I am unsure if this is valid. Any help on this would be much appreciated!
Background
I'm using bootstrapping to estimate if the respective population parameters for two populations $P_1$ and $P_2$ are significantly different.
For this purpose, I have samples from each population; namely, a sample $x_1$ of 900,000, and a sample $x_2$ of size 600,000. From these, I have done the following:

*

*I've computed test statistics $\bar{x_1}$ and $\bar{x_2}$¹, as well as their difference

*I've bootstrapped 2000 $x^∗_{1,i}$ 500,000-item samples and 2000 $x^{∗}_{2,i}$ 500,000-item samples

*I've computed 2000 test statistics $\overline{x^∗_{1,i}}$ and $\overline{x^∗_{2,i}}$, as well as the differences between each

¹These are not the means, but it is convenient to represent them this way. Anyone is free to change this if it is annoying.

Boostrap is used for assessing the uncertainty of the statistic. We know that the uncertainty changes with increasing sample size (approximately by $\sqrt n$), so if you use bootstrap samples that are smaller than the size of the data, you are overestimating the uncertainty. By using a smaller bootstrap sample size you can be almost certain that the result would be off, just how much off would depend on that particular scenario.
