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ON BOOTSTRAP ESTIMATION OF THE DISTRIBUTION

OF THE STUDENTIZED MEAN

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

^{1} *School of Mathematical Sciences, Centre for
Mathematics and
its Applications,*

Australian National University, Canberra, A.C.T. 0200,
Australia

^{2} *Department 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, self-normalization, Stable law,
Studentization.

**Source**
( TeX ,
DVI ,
PS )