I'm trying to put confidence intervals on parameters fitted through MLE through the inversion of the observed Fisher information matrix. More specifically, I define the observed FIM as:
$$ J_{n}(\hat{\theta{}_{n}}) = -\sum^{n}_{i=1}\frac{\partial{}^{2}}{\partial{}\hat{\theta{}_{n}^{2}}}\log{f_{\theta{}_{n}}(X_{i})} $$
Where $\log{f_{\theta{}}(X)}$ is the MLE term. I've seen in [some sources][1] that the confidence intervals for specific parameters can be estimated as (Equation 3.15 in source):
$$ \hat{\theta{}}_{nj} \pm{} c\sqrt{J_{n}(\hat{\theta_{n}})^{-1}_{jj}} $$
My confusion comes from accounting for the total number of data points. Sometimes it seems like these values are scaled by the number of data points, while others [seem to cancel out][2] (below equation 2 on page 2). A definitive answer and source would be really helpful.
Edit:
So to put it in more explicit, applied terms (might be conflating jargon throughout, so bear with me), we have an objective function like:
$$ l_{\theta{}}=\sum_{i=1}^{n}(-\frac{1}{2}\ln(2\pi{})-\frac{1}{2}\ln{\sigma_{i}^{2}}-\frac{1}{2}(\frac{\hat{y}_{i}(\theta{})-y_{i}}{\sigma_{i}})^{2}) $$
where $\theta{}$ is the vector of parameters being fitted, $n$ are the total number of data points in the set, $\sigma{}$ is the standard deviation of the noise around the experimental data, $\hat{y}_{i}(\theta{})$ is the model predicted outlet and $y$ is the measured outlet.
My goal is to minimize a function like this and to then have a covariance matrix around the fitted parameters (which should easily be converted into individual confidence intervals). Once I've reached the minimum, I can calculate the Hessian as:
$$ H_{\theta{}} = \frac{\partial{}^{2}l_{\theta{}}}{\partial{}\theta{}^{2}} $$
I've seen the covariance calculated through $H_{\theta{}}$ in two different ways: through the observed FIM and the expected FIM. If I were to choose the observed FIM, would it just be:
$$ V_{\theta} = H_{\theta}^{-1} $$
where $V_{\theta}$ is the covariance matrix. Would I be missing a scaling factor of $n$?
Then if I were to use the expected FIM, would I simply use:
$$ V_{\theta} = (E({H_{\theta}}))^{-1} $$
where $E$ is meant to be the expectation. [1]: https://www.stat.umn.edu/geyer/s06/5102/notes/fish.pdf [2]: https://arxiv.org/pdf/2107.04620.pdf