Metrika (1999) 50: 55±69 > Springer-Verlag 1999 Subsampling inference for the mean in the heavy-tailed case* Joseph P. Romano1, Michael Wolf2 1 Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA 94305, USA (e-mail: romano@stat.stanford.edu) 2 Departamento de Estadõ Âstica y Econometrõ Âa, Universidad Carlos III de Madrid, Calle Madrid 126, E-28903 Getafe, Spain (e-mail: mwolf@est-econ.uc3m.es) Received: December 1998 Abstract. In this article, asymptotic inference for the mean of i.i.d. ob- servations in the context of heavy-tailed distributions is discussed. While both the standard asymptotic method based on the normal approximation and Efron's bootstrap are inconsistent when the underlying distribution does not possess a second moment, we propose two approaches based on the sub- sampling idea of Politis and Romano (1994) which will give correct answers. The ®rst approach uses the fact that the sample mean, properly standardized, will under some regularity conditions have a limiting stable distribution. The second approach consists of subsampling the usual t-statistic and is somewhat more general. A simulation study compares the small sample performance of the two methods. Key words: Heavy tails, self-normalization, stable laws, subsampling 1 Introduction It has been two decades since Efron (1979) introduced the bootstrap proce- dure for estimating sampling distributions of statistics based on independent and identically distributed (i.i.d.) observations. While the bootstrap has en- joyed tremendous success and has led to something like a revolution of the ®eld 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 a < 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 * The ®nal version of this paper has bene®ted from helpful comments of two anonymous referees.