[R] Is this a mistake in 'An Introduction to R'?

[peter dalgaard]

On 04 Mar 2014, at 21:21 , Geoff Loveman <geoff at lovemans.co.uk> wrote:

> 
> 
> In 'An Introduction to R', section 11.7 on nonlinear least squares fitting,
> the following example is given for obtaining the standard errors of the
> estimated parameters:
> 
> "To obtain the approximate standard errors (SE) of the estimates we do:
> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))The 2 in the
> line above represents the number of parameters."
> 
> I know the inverted Hessian is multiplied by the mean square error and that
> the denominator of the MSE is the degrees of freedom (number of samples -
> number of parameters) but why does the numerator of the MSE (which is the
> RSS) get multiplied by the number of parameters? I have read through
> explanations of the method for obtaining the SE but I don't see where the
> MSE gets multiplied by the number of parameters or why this is needed as
> shown in the example?
> 


There are two 2's in that line, and I'd expect that only the last one has to do with the number of parameters, and the other one has to do with whether the Hessian is the second derivative of the sum of squares or of the negative loglikelihood function (half the sum of squares).

Quick check: In a linear model, we have

ssd = || Y- X beta ||^2
gradient = -2 (Y - X beta )'X
Hessian H = 2 X'X

and as we know, V(beta) = sigma^2 (X'X)^-1 = 2 sigma^2 H^-1

-pd

> Thanks for any help!
> 
> Geoff Loveman
> Tech lead SMERAS
> QQ Maritime Life Support
> 
> 
> 
> 
> 
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-- 
Peter Dalgaard, Professor
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com