Non-parametric bootstrap recycling

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 M.A. Newton and C.J. Geyer (J. Amer. Statist. Assoc. 89: 905–912, 1994). We show that it is bound to fail in non-parametric bootstrap applications, and suggest a modification that makes the method work. The modification consists of 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.