Distribution Estimation Using Concomitants of Order Statistics, with Application to Monte Carlo Simulation for the Bootstrap

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1992 · Volume 54 · Issue 2

DOI: 10.1111/j.2517-6161.1992.tb01899.x

Abstract

We show that a simulation method suggested by Efron for approximating bootstrap distributions is closely related to techniques based on concomitants of order statistics and develop its asymptotic properties from that viewpoint. We prove that the method produces Monte Carlo approximations with variance and mean-squared error decreasing likeB−1n−1/2, whereBdenotes the number of simulations andnequals the sample size. Therefore Efron's method can be a competitor with techniques such as balanced resampling, importance resampling and antithetic resampling, where variance and mean-squared error decrease likeB−1with no significant contribution from sample size. However, Efron's method has drawbacks, one being that performance in the tails is not as good as performance towards the centre of the distribution. In this respect, importance resampling has distinct advantages.

Keywords

bootstrap, concomitants of order statistics, distribution estimation, efficiency, monte carlo simulation, quantile, variance

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