I've fitted some weighted data to a nonlinear model (namely a Sèrsic light profile), using least squares.
That is, I have fitted a weighted nonlinear least squares model of the form:
$$I_r = I_0 \exp\left[-\left(\frac{r}{\alpha}\right)^{1/n}\right]+\epsilon_r$$
where $I_0$, $α$ and $n$ are parameters.
How do I then find confidence limits for each parameter, i.e. the 1,2,3 sigmas?
Best Answer
You can always fallback on bootstrapped confidence intervals.
Bootstrap resampling:
Let $X$ denote your training dataset. Let $n$ denote the number of samples in your training data. Let $k$ denote the number of resampling iterations you want to perform. The more the better, but $k$ should probably be no fewer than $1000$.
for $i=1,2,\dots k$, take a random sample $\tilde{X}_i$ (with replacement) of size $n$ from $X$. Train your model and calculate your model paramters. Let $\tilde{\theta}_i$ denote your fitted parameters trained on the $i^{th}$ resampled data set.
You can now calculate confidence intervals by determining the quantiles of $\tilde{\theta} = [\tilde{\theta}_1, \tilde{\theta}_2,\dots \tilde{\theta}_k]$. For example, to obtain a $95\%$ confidence interval, calculate the $2.5\%$ and $97.5\%$ quantiles of $\tilde{\theta}$.
For more on bootstrapping, reference chapters seven and eight of The Elements of Statistical Learning (available for free download).