Wild Bootstrap of the Sample Mean in the Infinite Variance Case

Abstract

It is well known that the standard independent, identically distributed (iid) bootstrap of the mean is inconsistent in a location model with infinite variance (α-stable) innovations. This occurs because the bootstrap distribution of a normalised sum of infinite variance random variables tends to a random distribution. Consistent bootstrap algorithms based on subsampling methods have been proposed but have the drawback that they deliver much wider confidence sets than those generated by the iid bootstrap owing to the fact that they eliminate the dependence of the bootstrap distribution on the sample extremes. In this paper we propose sufficient conditions that allow a simple modification of the bootstrap (Wu, 1986) to be consistent (in a conditional sense) yet to also reproduce the narrower confidence sets of the iid bootstrap. Numerical results demonstrate that our proposed bootstrap method works very well in practice delivering coverage rates very close to the nominal level and significantly narrower confidence sets than other consistent methods.

Keywords:

• Bootstrap
• Random probability measures
• Stable distributions
• Weak convergence

JEL Classification:

• C15

ACKNOWLEDGMENTS

We are grateful to two anonymous referees, an Associate Editor and the Editor for their helpful and constructive comments on a previous version of the paper. Cavaliere and Georgiev acknowledge the financial support of the Portuguese Fundaço para a Ci\^encia e a Tecnologia. Project PTOC/EGE-ECO/108620/2008.

Notes

This can be seen by noting that can be written as the partial sum of the waiting times of the Poisson process, which are iid Exp(1).

We are very grateful to one of the referees for pointing this out to us.

^a These are the quartiles of the ratio between the length of the bootstrap confidence sets and the length of an exact (unconditional) confidence set.