I would like to use this procedure for estimation of the confidence interval in Nonlinear Least Squares:
How to estimate confidence interval of a least-squares fit parameters by means of numerical Jacobian
However my error is weighted:
$$ \epsilon_n = (w_n(y_n-f(x_n))$$
Most of time I use a relative error, weighting by the inverse of the data during fitting, because my data cover several order of magnitude:
$$\epsilon_n = (\frac{y_n-f(x_n)}{y_n})$$
How should I calculate the confidence intervals in both cases?
According to this, which is valid for linear LS I would suggest for NLS that:
\begin{eqnarray*} Var\left(\hat{\boldsymbol{\beta}}_{weighted}\right) & = & \sigma^{2}\left(\boldsymbol{J}^{\prime}\boldsymbol{WJ}\right)^{-1} \end{eqnarray*}
were J is the jacobian and $\sigma$ the variance
The 3th parameter + CI would be: \begin{align} \beta_{3 \ weighted} \pm 1.96\sqrt{Var\left(\hat{\boldsymbol{\beta}}_{weighted} \right)_{33}} \end{align}
Is it true?
Note: According to Chapter 11 of Applied Linear Statistical Models by Kutner, Nachtsheim, Neter, and Li, 5th ed. : \begin{eqnarray*} \hat{\sigma}_r^2=\frac{1}{N-k}\sum_{n=1}^N\left(\epsilon_n)\right)^2 \end{eqnarray*}
Regards
