  Bootstrapping was designed to randomly resample data from a fixed sample
using Monte Carlo techniques. However, the original sample itself defines a
discrete distribution. Convolutional methods are well suited for discrete
distributions, and we show the advantages of utilizing these techniques for
bootstrapping. The discrete convolutional approach can provide exact numerical
solutions for bootstrap quantities, or at least mathematical error bounds. In
contrast, Monte Carlo bootstrap methods can only provide confidence intervals
which converge slowly. Additionally, for some problems the computation time of
the convolutional approach can be dramatically less than that of Monte Carlo
resampling. This article provides several examples of bootstrapping using the
proposed convolutional technique and compares the results to those of the Monte
Carlo bootstrap, and to those of the competing saddlepoint method.
