Bootstrapping the Sample Mean for Data with Infinite Variance.

摘要

When data comes from a distribution belonging to the domain of attraction of a stable law, Athreya (1987a) showed that the bootstrapped sample mean has random limiting distribution, implying that the naive bootstrap could fail in the heavy-tailed case. The goal here is to classify all possible limiting distributions of the bootstrapped sample mean when the sample comes from a distribution with infinite variance, allowing the broadest possible setting for the (nonrandom) scaling, the resample size, and the mode of convergence (in law). The limiting distributions turn out to be infinitely divisible with possibly random Levy measure, depending on the resample size. An averaged-bootstrap algorithm is then introduced which eliminates any randomness in the limiting distribution. Finally, it is shown that (on the average) the limiting distribution in the domain of (partial) attraction of a stable law. (Author) (kr)