Ma. Newton et Cj. Geyer, BOOTSTRAP RECYCLING - A MONTE-CARLO ALTERNATIVE TO THE NESTED BOOTSTRAP, Journal of the American Statistical Association, 89(427), 1994, pp. 905-912
A Monte Carlo algorithm is described that can be used in place of the
nested bootstrap. It is particularly advantageous when there is a prem
ium on the number of bootstrap samples, either because samples are har
d to generate or because expensive computations are applied to each sa
mple. This recycling algorithm is useful because it enables inference
procedures like prepivoting and bootstrap iteration in models where ne
sted bootstrapping is computationally impractical. Implementation of t
he recycling algorithm is quite straightforward. As a replacement of t
he double bootstrap, for example, bootstrap recycling involves two sta
ges 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 measur
e; a measure dominating all the first-stage fits. These samples are re
cycled 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 con
tingency tables and to construction of likelihood-based confidence set
s in a hidden Markov model from hematology.