Richardson Extrapolation and the Bootstrap
提出用理查森外推法降低自助法模拟的计算成本，通过两个较小样本量的模拟来近似大样本结果，并理论研究了最佳样本量比例。
Abstract Simulation methods [particularly Efron's (1979) bootstrap] are being applied more and more frequently in statistical inference. Given data (X 1 …, Xn ) distributed according to P, which belongs to a hypothesized model P, the basic goal is to estimate the distribution L P of a function Tn (X 1, …, Xn, P). The bootstrap presupposes the existence of an estimate and involves estimating L P by the distribution L* n of ), where (X 1*, …, Xn *) is distributed according to The method is of particular interest when Ln *, though known in principle, can realistically only be computed by simulation. Such computation can be expensive if n is large and Tn is complex (e.g., see the multivariate goodness-of-fit tests of Beran and Millar 1986). Even when bootstrap application to a single data set is not excessively expensive, Monte Carlo studies of the bootstrap are another matter. We propose a method based on the classical ideas of Richardson extrapolation for reducing the computational cost inherent in bootstrap simulations and Monte Carlo studies of the bootstrap, by performing simulations for statistics based on two smaller sample sizes. We study theoretically which ratio of the two small sample sizes is apt to give best results. We show how our method works for approximating the χ2, t, and smoothed binomial distributions, and for setting bootstrap percentile confidence intervals for the variance of a normal distribution with a mean of 0.