ON BOOTSTRAP ESTIMATION OF THE DISTRIBUTION OF THE STUDENTIZED MEAN

PETER HALL^{1} AND RAOUL LEPAGE^{1,2}

1School of Mathematical Sciences, Centre for Mathematics and its Applications,

Australian National University, Canberra, A.C.T. 0200, Australia 2Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, U.S.A.

(Received July 29, 1994; revised October 20, 1995)

Abstract. It is shown that bootstrap methods for estimating the distribution of the Studentized mean produce consistent estimators in quite general contexts, demanding not a lot more than existence of finite mean. In particular, neither the sample mean (suitably normalized) nor the Studentized mean need converge in distribution. It is unnecessary to assume that the sampling distribution is in the domain of attraction of any limit law.

Key words and phrases: Bootstrap, central limit theorem, consistency, domain of attraction, domain of partial attraction, heavy tail, percentile-t method, selfnormalization, Stable law, Studentization.