An alternative to the mm out of nn bootstrap

It is well known that Efron's bootstrap can fail in settings where the data are heavy tailed and when regularity conditions do not hold. Naturally this applies to weighted bootstrap schemes such as the Bayesian bootstrap. To deal with this, we introduce a Bayesian bootstrap analogue of the mm out of nn bootstrap. This bootstrap differs from traditional mm out of nn bootstraps in that all nn observations are used in the bootstrap test statistic. Moreover, the method is relatively robust to the selection of mm. We establish consistency for the new bootstrap and examine its other useful properties including a connection to the Dirichlet process. Several examples illustrating consistency in settings where the Efron bootstrap fails are given. Further generalizations are suggested.