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$\DeclareMathOperator{\var}{Var}$ How to compute prediction bands for non-linear regression?

In the above link, you have mentioned about the variance-covariance matrix of the estimates. What is the explicit formula for computing the terms of this variance-covariance matrix?

If suppose, $f = f(x;p1,p2,p3)$ with $x$ as the independent variable and $P=(p1,p2,p3)$ are the nonlinear parameters of a function which fits the data in a nonlinear regression problem, what is $\var(P)$?

Is the Hessian equal to the one presented in this link? https://www8.cs.umu.se/kurser/5DA001/HT07/lectures/lsq-handouts.pdf

or

http://www.math.vt.edu/people/dlr/m2k_svb11_hesian.pdf

Or, are they the same? Can you explain the difference?

Do the inverse of the above Hessians mentioned in the above links estimate the variance-covariance matrix?

Which one of the Hessian formula must be taken into consideration for evaluating the Hessian you meant in the first link mentioned above?

Rizwan

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    $\begingroup$ This post will help. see espeically the answer and the different relationship between Hessian and var-cov depending on the objective function. $\endgroup$
    – qoheleth
    Sep 12, 2014 at 4:10

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