Resampling and exchangeable arrays

The nonparametric, or resampling, bootstrap for a single unstructured sampl e corresponds to the algebraic operation of monoid composition, with a unif orm distribution on the monoid. With this interpretation, the notion of res ampling can be extended to designs having a certain group-invariance proper ty. Two types of exchangeable array structures are considered in some detai l, namely the one-way layout, and the two-way row-column exchangeable desig n. Although in both cases there is a unique group under which the sampling distribution of the observations is exchangeable, the choice of monoid is n ot unique. Different choices of monoid can lead to drastically different, a nd in some cases quite misleading, inferences.