[R] Standard error of standard deviation: bootstrap or theoretical results?

[Prof Brian Ripley]

On Wed, 6 Aug 2003, Thomas W Blackwell wrote:

> Huan  -
> 
> The difference between the empirical ("bootstrap') result and the
> theoretical results shows evidence for autocorrelation in the time
> series data.

I don't think that's where the correlation is (and for positive 
autocorrelation I would expect 4 to be larger than 1 to 3).

> -  tom blackwell  -  u michigan medical school  -  ann arbor  -
> 
> On Wed, 6 Aug 2003 huan.huang at bnpparibas.com wrote:
> 
> > This is more a statistical question rather than an R question. I'd
> > appreciate it if you can give me some suggestions.

Suggestion: R-help is not a source of free statistical consultancy.
Please seek expert advice within your company, or employ a qualified 
consultant.  Which is why I am only pointing out the most obvious mistakes 
here, and not proffering possible solutions.

> > I have a sample of a time series (sample size 500, fat tail in density). I
> > am trying to calculate the Standard error of standard deviation of a
> > sub-block-sample (sample size 250). I take 100 this kind of
> > sub-block-sample, randomly. For these 100 subsamples, I use the following 3
> > methods to calculate the standard error of standard deviation:

But those 100 subsamples cannot be independent.  If you take a blocks of 
size 250 out of 500, they are almost bound to overlap.

> > 1. From book "Handbook of applicable mathematics", Walter Ledermann (chief
> > editor) Volumn VI: Statistics, Part A, Lloyd, John Wiley & Sons.
> >
> > Page 30-32:
> >
> > var(S) = (mu4 - mu2^2)/(4 * mu2 * n)
> > mu4 = E(X - mu)^4, mu2 = E(X - mu)^2, S^2 = sum(X - mu)^2/n
> >
> > The results are about: 0.00090

That is assuming independent samples.

> > 2. From   http://davidmlane.com/hyperstat/A19196.html
> > The results are about 0.00066

That's for iid *normal* samples.

> > 3. From   http://mathworld.wolfram.com/StandardDeviationDistribution.html
> > The results are about 0.00065

Same.

> > Finally I calculate the standard deviation for each of the 100 subsamples
> > and the standard error of those 100 standard deviations ( I reckon this is
> > the bootstrap result for the standard error of the standard deviation I
> > want).

That is not a bootstrap, at least not in any valid sense.  You can 
bootstrap time series, but it is tricky to find a valid method.

> > I get 0.00024
> >
> > I tried all above a couple of times and got similar results for each
> > methods I used. The results from the first 3 methods are apparently higher
> > than the bootstrap one. I am a bit confused. Do I miss anything? Which one
> > do you believe?

You seem to have missed the need to check your assumptions, none.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595