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I want to fit a function $f(x_1,x_2..)$ to (noisy) data with unknown variance. For each datapoint, I have a weight $w_i$ which is proportional to the reliability of that particular datapoint. The real uncertainty of the datapoints is unknown. I do the fitting using Matlab and Levenberg-Marquardt so that I end up with the parameter estimates $\hat{\beta}$ and the Jacobian $J$. I calculate the variance of the parameter estimates as follows: $$\hat{\sigma}^2=\frac{RSS}{n-p+1} \cdot (J'WJ)^{-1}$$ Where $W$ is the diagonal matrix of the weights, $n$ is the number of datapoints and $p$ is the degrees of freedom.

My question is regarding the normalization of the weights and it's effect on the parameter uncertainty, since I have seen some conflicting definitions and opinions on this subject.

Question: How do I normalize my weights so that the calculated parameter standard errors are meaningful? I currently normalize the weights so that their sum is the number of datapoints $N$, similar to non-weighted least squares: $\Sigma w_i = N$. Is this correct?

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