Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 3, 640--652



PETER J. BICKEL University of California, Berkeley, USA
ANAT SAKOV Technion, Israel Institute of Technology,Haifa, Israel

SUMMARY. The $m$ out of $n$ bootstrap, with or without replacement, where $m \rightarrow \infty$ and $m/n \rightarrow 0$ has been proposed on two grounds: (i) As a way of ensuring consistency when the classical bootstrap is not consistent. (ii) When it is consistent, then in conjunction with extrapolation, as a way of obtaining behaviour equivalent to that of the classical bootstrap, to second or higher order, with reduced computation time. In this paper we shall discuss a partial taxonomy of higher order behaviour of the $m$ out of $n$ bootstrap and introduce a general form of extrapolation.

AMS (1991) subject classification}. Primary 62G09; secondary 62G05.

Key words and phrases. Bootstrap, extrapolation, subsampling, Edgeworth expansion.

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