673

**Nonparametric Bootstrap Recycling**

**Valérie Ventura**

Revised 12/00

### Abstract:

The double bootstrap provides diagnostics for bootstrap
calculations and, if need be, appropriate adjustments. The
amount of computation involved is usually considerable, and
recycling provides a less computer intensive alternative. Recycling
consists of using repeatedly the same samples drawn from a recycling
distribution *G* for estimation under each first-level bootstrap
distribution, rather than independently repeating the simulation
and estimation steps for each of these.

Recycling is successful in parametric applications of the bootstrap, as demonstrated
by Newton and Geyer (1994). We show that it is bound to fail in nonparametric bootstrap
applications, and suggest a modification that makes the method work. The modification consists in smoothing the first-level bootstrap distributions, with the desired
consequence that this removes the zero probabilities in the
multinomial
distributions that define them.
We also discuss efficient choices of recycling distributions, both in
terms of estimator efficiency and simulation efficiency.

*Keywords:* Bootstrap Adjustments, Bootstrap Diagnostics, Double Bootstrap, Frequency smoothing, Importance Sampling, Pseudobias.

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