Sebastien,
> Hi Dimitri,
> I'm obviously missing something in my litterature review. I did a new
> MC simulation, with a much smaller number of observation points
> (namely 3, to fit a straight line!!!). It turns out that the formula
> you are advocating for is the best estimate of the standard deviation
> of the parameters. Could you please explain why this fomula differs
> from formulas (34) and (35) in
> http://mathworld.wolfram.com/LeastSquaresFitting.html?
First thing worth noting is Worlfram is wise enough to call 34 and 35
standard error ... and not standard deviation!
As Gilles and you have shown with your MC simulations, the standard
deviation (sigma_i=sqrt(cov[i][i])) approximates by how much the fitted
parameter can vary when several sets of 'observations' are sampled with
the same error distribution. I wrote 'approximate' because the true
standard deviation is not accessible, instead it is approximated as the
inverse of Fisher information matrix which is directly related to the
Hessian matrix. The relation between Fisher and the variance of the
parameter is known as the Rao-Cramer bound.
In the case of the standard error, the sample of observations is fixed
and one wonders by how much one can change the parameters without
changing the resulting normalized chi square too much. That is the
role of s (eq. 32 on Wolfram). It should be noted that nowhere on
that page there is the notion of error on the observations: the data
are what they are and no alternative sampling should be considered.
Please, have a look at
http://en.wikipedia.org/wiki/Standard_deviation
http://en.wikipedia.org/wiki/Standard_error
for further details, especially the last section of the Standard_error
page as it compares std. error and deviation.
Regards,
Dim.
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Dimitri Pourbaix * Don't worry, be happy
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