Tail-Specific Linear Approximations for Efficient Bootstrap Simulations

Abstract

Two effective variance-reduction techniques for estimating probabilities and quantiles in the tails of bootstrap distributions—importance sampling and concomitants of order statistics—are based on linear approximations. Although these techniques offer potential asymptotic variance reductions by factors of nine to infinity, in practice the reductions may be only by a factor of two or smaller because of inaccurate linear approximations. We develop tail-specific linear approximations that are more accurate where the accuracy is important, in the tails of distributions. Our methods fall into two categories—influence function methods and regression methods. Both can be applied without problem-specific analytical calculations, and both have tail-specific versions. We apply the tail-specific approximations to importance sampling and concomitants and propose another technique that uses linear approximations, post-stratification implemented using the saddlepoint. This technique shares the same O(n ^{-1/2 B -1}) variance as the concomitants procedure. Tail-specific approximations improve the performance of the variance-reduction techniques by a factor of about three in our simulations.

Key Words:

• Concomitants of order statistics
• Importance sampling
• Jackknife
• Stratified sampling
• Variance reduction