Bootstrap recycling: A monte carlo alternative to the nested bootstrap

A Monte Carlo algorithm is described that can be used in place of the nested bootstrap. It is particularly advantageous when there is a premium on the number of bootstrap samples, either because samples are hard to generate or because expensive computations are applied to each sample. This recycling algorithm is useful because it enables inference procedures like prepivoting and bootstrap iteration in models where nested bootstrapping is computationally impractical. Implementation of the recycling algorithm is quite straightforward. As a replacement of the double bootstrap, for example, bootstrap recycling involves two stages of sampling, as does the double bootstrap. The first stage of both algorithms is the same: simulate from the fitted model. In the second stage of recycling, one batch of samples is simulated from one measure; a measure dominating all the first-stage fits. These samples are recycled with each first-stage sample to yield estimated adjustments to the original inference procedure. Choice of this second-stage measure affects the efficiency of the recycling algorithm. Gains in efficiency are slight for the nonparametric bootstrap but can be substantial in parametric problems. Applications are given to testing with sparse contingency tables and to construction of likelihood-based confidence sets in a hidden Markov model from hematology.