An alternative to the out of bootstrap

It is well known that Efronʹs 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 m out of n bootstrap. This bootstrap differs from traditional m out of n bootstraps in that all n observations are used in the bootstrap test statistic. Moreover, the method is relatively robust to the selection of m . 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.