Subsampling Inference with K Populations and a Non‐standard Behrens–Fisher Problem

We revisit the methodology and historical development of subsampling, and then explore in detail its use in hypothesis testing, an area which has received surprisingly modest attention. In particular, the general set‐up of a possibly high‐dimensional parameter with data from K populations is explored. The role of centring the subsampling distribution is highlighted, and it is shown that hypothesis testing with a data‐centred subsampling distribution is more powerful. In addition we demonstrate subsampling’s ability to handle a non‐standard Behrens–Fisher problem, i.e., a comparison of the means of two or more populations which may possess not only different and possibly infinite variances, but may also possess different distributions. However, our formulation is general, permitting even functional data and/or statistics. Finally, we provide theory for K ‐ sample U ‐ statistics that helps establish the asymptotic validity of subsampling confidence intervals and tests in this very general setting.