# How to estimate confidence intervals of an arbitrarily weighted least-squares fit parameters

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