Fast Bootstrapping by Combining Importance Sampling and Concomitants

Importance sampling is the old standby method for ob taining accurate tail quantiles of a bootstrap distribution more quickly A newer method a variation of control variates called concomitants is especially attractive in larger problems because its e ciency relative to simple Monte Carlo sampling increases at the rate of p n where n is the sample size We show how to combine these complementary meth ods Doing so successfully requires two modi cations to classical importance sampling a weighted average esti mate and a mixture design distribution and the use of saddlepoint estimates for the concomitants These meth ods can be programmed to run automatically and o er improved moment estimation simultaneous with quan tile estimation The e ciency gains in can be large e g by a factor of even with small n We also obtain promising results by smoothing the distribution estimates produced by concomitants with and without importance sampling