I have two questions on applying the bootstrap. Neither of these appear to be
covered in Efron and Tibshirani's excellent introductory book (1993) 'an introduction
to the bootstrap'.
Suppose the original sample S contains N elements and comes from a population
P.
(1) Is there a minimum value for N for which bootstrap resampling (in other
words, construction of resamples of size N by randomly sampling from S) is either
NOT valid or not likely to be a good method for estimating the standard error
or calculating a percentile-based confidence interval of a given statistic of
interest for P?
(2) If resamples of size M are constructed from S WHERE M IS STRICTLY GREATER
THAN N, then:
(a) as M increases, the bootstrap resamples (of size M) generated from S (of
size N) become 'less representative' of the original population P and thus any
estimate of the standard error or a confidence interval calculated for a chosen
statistic of P is
prone to increasing bias as M increases,
(b) if the statistic of interest is the mean Mu of P, then the distribution
of the 'resample mean' as calculated from the bootstrap resamples generated
from S tends towards a NORMAL distribution centred on Mu.
If there are any references which cover these points I would appreciate hearing
of them.
Many thanks,
Eric Grist
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