Subsampling the Mean with Heavy Tails

It has been two decades since Efron (1979) introduced the bootstrap procedure for estimating sampling distributions of statistics based on independent and identically distributed (i.i.d.) observations. While the bootstrap has enjoyed tremendous success and has led to something like a revolution in the field of statistics, it is known to fail for a number of counterexamples. One well-known example is the case of the mean when the observations are heavy-tailed. If the observations are i.i.d. according to a distribution in the domain of attraction of a stable law with index α < 2 (see Feller, 1971), then the sample mean, appropriately normalized, converges to a stable law. However, Athreya (1987) showed that the bootstrap version of the normalized mean has a limiting random distribution, implying inconsistency of the bootstrap. An alternative proof of Athreya’s result was presented by Knight (1989). Kinateder (1992) gave an invariance principle for symmetric heavy-tailed observations. It has been realized that taking a smaller bootstrap sample size can result in consistency of the bootstrap, but knowledge of the tail index of the limiting law is needed (see Section 1.3 and Athreya, 1985, and Arcones, 1990; also see Wu, Carlstein, and Cambanis, 1993, and Arcones and Giné, 1989).