Do both Bootstrap with and without replacement create a distribution?

Question

I'm having a "noisy debate" with colleagues about whether sampling without replacement can still create a distribution.

Methodology: A bootstrap (iterative process where I calculate Somers' D for new samples) is done with and without replacement.

I am sampling without replacement on the first level of my primary key, so each time I subsample, I take 80% of my full population and with each first level (name of client) I can get X observations, where X is not static)

Dispute: While I would like to do both types of bootstrap, a loop with and without replacement - apparently, sampling without replacement, over 1,000 iterations and saving the Somers' D does not create a distribution of Somers' D?

Resolution: Please could someone let me know if I am correct or incorrect in saying "In almost any case, when you repeatedly take a smaller sample of your larger sample where each smaller sample is independently and identically distributed (the subsample is taken in an identical process but taken independently from another) then your Somers' D results can and will create a distribution?

Thanks!

Answer 1

Bootstrap methods require resampling using the same sample size as that of the original sample. If you use fewer (or more!) samples, this is called "upstrap"

The Upstrap.

This, however, has different properties and you should be careful as otherwise the precision of your estimates would be misleading.