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This may be a lower level question, so thank you in advance to any patient person who has an answer for me. I develop non-linear mixed effects models of pharmaceutical kinetics. One tool I'm frequently using is the observed Fisher information matrix. As I understand, the standard definition is,

$$ \mathbf{I}(\theta)=-\frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}l(\theta),~~~~ 1\leq i, j\leq p $$

What I use it most frequently for is deriving the standard error and variance of parameter estimates via,

$$ \mathrm{Var}(\hat{\theta}_{\mathrm{ML}})=[\mathbf{I}(\hat{\theta}_{\mathrm{ML}})]^{-1} $$

However, I am unable to find a derivation which explains why this works :/ Can anyone explain to me why the inverse of the Fisher information matrix is an estimate of the variance-covariance matrix for the population parameters?

Thanks so much m-_-m

EDIT: I found the answer I wanted using the following document

BennyBoi
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  • Perhaps it will help to modify your characterization a little, but in an important way: the population *parameters* have no variance-covariance matrix, because they aren't random variables. It is their *estimates* which, by virtue of being functions of the data, are random variables and those have a variance-covariance matrix. – whuber Sep 30 '20 at 14:44
  • @usεr11852, I read through the reply you sent me. Honestly, it doesn't make much sense to me :c It just leaves me with more questions. Now I'm wondering what the heck Fisher information even is. – BennyBoi Sep 30 '20 at 16:03
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    @SimonWood's textbook "core statistics" (freely available online) has a derivation of this result. – Jarle Tufto Sep 30 '20 at 19:56

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