SMOOTHING THE BOOTSTRAP

The question of smoothing when using the non-parametric version of the bootstrap for estimation of population functionals is reconsidered. In general, there is no global preference for procedures based on a smoothed version of the empirical distribution rather than the empirical distribution itself. In the majority of problems smoothing influences only the second order properties of the estimator, while requiring greater computation and choice of a suitable amount of smoothing. There are problems, however, where smoothing may affect the rate of convergence of the estimator. We discuss an example of such a problem and consider issues relating to empirical choice of whether to smooth, and by how much. A procedure based on the bootstrap for choice of bandwidth is suggested and illustrated.