On the relative performance of bootstrap and Edgeworth approximations of a distribution function

Performance of the bootstrap for estimating tail probabilities is usually explained by saying that the bootstrap provides a one-term Edgeworth correction. However, simulation studies show that the bootstrap usually performs better than explicit Edgeworth correction. We present a theory which explains this empirical observation. The theory is based on a comparison of relative error in bootstrap and Edgeworth approximation formulae and uses expansions of large deviation probabilities. We treat general Edgeworth approximations, not simply the one-term corrections usually associated with the bootstrap. We show that bootstrap and Edgeworth approximations are equivalent out to a certain distance in the tail. Beyond that point the bootstrap performs markedly better than Edgeworth correction, except for the case of extreme tail probabilities where it is possible for bootstrap and Edgeworth approximations to outperform one another, depending on the sign of skewness. In the case of one-term Edgeworth correction the bootstrap performs markedly better for both moderate and large deviations, except in the extreme tails. Even there the bootstrap outperforms Edgeworth correction if skewness is of the right sign.