On Wed, Nov 3, 2010 at 3:54 PM, Henrik Bengtsson <h...@biostat.ucsf.edu>wrote:
> Hi, consider this one as an FYI, or a seed for further discussion.
>
> I am aware that many traps on sample() have been reported over the
> years. I know that these are also documents in help("sample"). Still
> I got bitten by this while writing
>...
> All of the above makes sense when one study the code of sample(), but
> sample() is indeed dangerous, e.g. imagine how many bootstrap
> estimates out there quietly gets incorrect.
Nonparametric bootstrapping from a sample of size 1 is <always> incorrect.
If you draw a single observation from a sample of size 1, you get that
same observation back. This implies zero sampling variability, which
is wrong. If this single sample represents one stratum or sample in
a larger problem, this would contribute zero variability to the overall
result, again wrong.
In general, the ordinary bootstrap underestimates variability in
small samples. For a sample mean, the ordinary bootstrap corresponds
to using an estimate of variance equal to (1/n) sum((x - mean(x))^2),
instead of a divisor of n-1. In stratified and multi-sample applications
the downward bias is similarly (n-1)/n.
Three remedies are:
* draw bootstrap samples of size n-1
* "bootknife" sampling - omit one observation (a jackknife sample), then
draw a bootstrap sample of size n from that
* bootstrap from a kernel density estimate, with kernel covariance equal
to empirical covariance (with divisor n-1) / n.
The latter two are described in
Hesterberg, Tim C. (2004), Unbiasing the Bootstrap-Bootknife Sampling vs.
Smoothing, Proceedings of the Section on Statistics and the Environment,
American Statistical Association, 2924-2930.
http://home.comcast.net/~timhesterberg/articles/JSM04-bootknife.pdf
All three are undefined for samples of size 1. You need to go to some
other bootstrap, e.g. a parametric bootstrap with variability estimated
from other data.
Tim Hesterberg
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