Bootstrap relative errors and sub-exponential distributions

Abstract

For the purposes of this paper, a distribution is sub-exponential if it has finite variance but its moment generating function is infinite on at least one side of the origin. The principal aim here is to study the relative error properties of the bootstrap approximation to the true distribution function of the sample mean in the important sub-exponential cases. Our results provide a fairly general description of how the bootstrap approximation breaks down in the tail when the underlying distribution is sub-exponential and satisfies some very mild additional conditions. The broad conclusion we draw is that the accuracy of the bootstrap approximation in the tail depends, in a rather sensitive way, on the precise tail behaviour of the underlying distribution. Our results are applied to several sub-exponential distributions, including the lognormal. The lognormal case is of particular interest because, as the simulation studies of Lee and Young have shown, bootstrap confidence intervals can have very poor coverage accuracy when applied to data from the lognormal.