🚀 go-pugleaf

news@massey.ac.nz (USENET News System) writes:

>In article <1992Aug16.212245.27577@mailhost.ocs.mq.edu.au>, wskelly@laurel.ocs.mq.edu.au (William Skelly) writes:
>> 
>> This and other posting indicate that there is a relationship between
>> sample size and and estimated variance (of the population) which is
>> positive and always an underestimate.  What is the limit, or point
>> at which an increasing sample size no longer improve the estimate
>> of populations variance?
>> 
>When the sample size is equal to the population size (never for an
>infinite population).

Not necessarily true.  It depends on the sample design.  Suppose that
we have a finite population of size N and we draw _with_replacement_ a
sample of size n using independent draws.  Then the bias from using
the "n" denominator is (minus)

          <population variance> / n

and this is true for _any_ n, including n = N, or even n = 2N.

>> Can this be tested by taking samples of sample 

>Yes but we can work out the theory so it isn't necessary.

>> (the later sample
>> being elevated to the status of population)?
>> 

Perhaps we can work out the theory for the bias, but this is not so
clear when we consider other features of the distribution of the
estimate.  See the related questions about the bias of the usual
SD estimate in this thread for an example.

The intuition in the original comment was right on!

In fact the procedure of taking samples from the original sample,
"elevating" the original sample to the status of population is
_exactly_ the definition of (a particular variety of) bootstrap
resampling.  See for example the book _The_Bootstrap_and_Edgeworth_
_Expansion_, by Peter Hall (Springer-Verlag, 1992), which provides
many references and examples.

It is shown in the bootstrap literature that for certain purposes
(e.g. constructing statistical tests when the sample is drawn from a
non-normal population) the bootstrap procedure can outperform
(sometimes significantly) the traditional procedures.

So, for example, if the reason you are estimating a variance is to put
its square root in the denomenator of a t-statistic, then boostrapping
_may_ be the way to go (unless you really believe in the normality of
your population).  For this application, you really aren't interested
in whether or not your estimate of the variance is biased, and so the
n vs. n-1 debate is irrelevant.  In fact, the bootstrap based test
procedure will lead to the same test whether you choose n _or_ n-1, an
invariance property that I find appealing.

T. Scott Thompson              thompson@atlas.socsci.umn.edu
Dept. of Economics
University of Minnesota