On a Set of the First Category

In an analysis of the bootstrap Putter & van Zwet (1993) showed that under quite general circumstances, the bootstrap will work for “most” underlying distributions. In fact, the set of exceptional distributions for which the bootstrap does not work was shown to be a set D of the first category in the space S of all possible underlying distributions, equipped with a topology S. Such a set of the first category is usually “small” in a topological sense. However, it is known that this concept of smallness may sometimes be deceptive and in unpleasant cases such “small” sets may in fact be quite large. Here we present a striking and hopefully amusing example of this phenomenon, where the “small” subset D equals all of S. We show that as a result, a particular version of the bootstrap for the sample minimum will never work, even though our earlier results tell us that it can only fail for a “small” subset of underlying distributions. We also show that when we change the topology on S—and as a consequence employ a different resampling distribution—this paradox vanishes and a satisfactory version of the bootstrap is obtained. This demonstrates the importance of a proper choice of the resampling distribution when using the bootstrap.