Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples. We use two innovations in control variates-interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n(-1/2)B(-1), where n is the sample size and B is the bootstrap sample size. We discuss two modifications to classical importance sampling-a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.