R help - Feb 2004 - Obtaining SE from the hessian matrix

Dear R experts,

In R-intro, under the 'Nonlinear least squares and maximum likelihood
models' there are ttwo examples considered how to use 'nlm'
function.
In 'Least squares' the Standard Errors obtained as follows:

    After the fitting, out$minimum is the SSE, and out$estimates are the
    least squares estimates of the parameters. To obtain the approximate
    standard errors (SE) of the estimates we do:

    > sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))

But under 'Maximum likelihood' section I've read:

    After the fitting, out$minimum is the negative log-likelihood, and
    out$estimates are the maximum likelihood estimates of the parameters.
    To obtain the approximate SEs of the estimates we do:

    > sqrt(diag(solve(out$hessian)))

As for me, I use MINPACK fortran library for NLS fitting in R, and there
I also get the hessian matrix. What formula should I use in _this_ case?
Well, some times ago I had a glance at gsl, GNU Scientific Library. It
use converted-to-C MINPACK for NLS fit too.  And, in the GSL ref. manual
example
    http://www.gnu.org/software/gsl/manual/html_node/gsl-ref_36.html#SEC475
SE calculated as
    #define ERR(i) sqrt(gsl_matrix_get(covar,i,i))

where covar = (J^T * J)^-1 (i.e. how in the second formulae above).
So, what is the _right_ way for obtatining SE? Why two those formulas above
differ?

Thank you!

--
WBR,
Timur.

On Thu, 19 Feb 2004, Timur Elzhov wrote:
> So, what is the _right_ way for obtatining SE? Why two those formulas above
> differ?
>
If you are maximising a likelihood then the covariance matrix of the
estimates is (asymptotically) the inverse of the negative of the Hessian.

The standard errors are the square roots of the diagonal elements of the
covariance.

So if you have the Hessian you need to invert it, if you have the
covariance matrix, you don't.

	-thomas