Resampling and exchangeable arrays

The nonparametric, or resampling, bootstrap for a single unstructured sample corresponds to the algebraic operation of monoid composition, with a uniform distribution on the monoid. With this interpretation, the notion of resampling can be extended to designs having a certain group-invariance property. Two types of exchangeable array structures are considered in some detail, namely the one-way layout, and the two-way row-column exchangeable design. Although in both cases there is a unique group under which the sampling distribution of the observations is exchangeable, the choice of monoid is not unique. Different choices of monoid can lead to drastically different, and in some cases quite misleading, inferences.