Bootstrap recycling: A Monte Carlo alternative to the nested bootstrap

### Bootstrap recycling: A Monte Carlo alternative to the nested bootstrap

Michael A. Newton
and Charles J. Geyer

1994 *Journal of the American Statistical Association*, **89**, 905-912.

### Abstract:

A Monte Carlo algorithm is described which 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 {\it 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 {\it 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 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.

**Keywords:** nested bootstrap, prepivot, bootstrap diagnostics, Monte Carlo,
composite hypothesis