Speeding up bootstrap computations: a vectorized implementation for statistics based on sample moments

In this note we propose a vectorized implementation of the non-parametric bootstrap for statistics based on sample moments. Basically, we adopt the multinomial sampling formulation of the non-parametric bootstrap, and compute bootstrap replications of sample moment statistics by simply weighting the observed data according to multinomial counts, instead of evaluating the statistic on a re-sampled version of the observed data. Using this formulation we can generate a matrix of bootstrap weights and compute the entire vector of bootstrap replications with a few matrix multiplications. Vectorization is particularly important for matrix-oriented programming languages such as R, where matrix/vector calculations tend to be faster than scalar operations implemented in a loop. We illustrate the gain in computational speed achieved by the vectorized implementation in real and simulated data sets, when bootstrapping Pearson's sample correlation coefficient.