# Practical application of Cramer-Rao lower bound to calculate the variance of estimator

I would like to use the Cramer-Rao lower bound to help me estimate the variance of my maximum likelihood estimator, across a range of different samples of data.

My question is, how do I do this correctly? My model has two parameters, $\mu$ and $\psi$.

Thus far I have been evaluating the following at the maximum likelihood estimates of my two parameters $\hat{\mu}$ and $\hat{\psi}$:

$I_{11}=\frac{\partial^2 ln L(\mu,\psi)}{\partial \mu^2}$,$I_{12}=I_{21}=\frac{\partial^2 ln L(\mu,\psi)}{\partial \mu \partial \psi}$,$I_{22}=\frac{\partial^2 ln L(\mu,\psi)}{\partial \psi^2}$

I then put these into a matrix, making a 2x2 information matrix of the form: $I_{estimate}=[[I_{11} ,I_{12}],[I_{21} ,I_{22}]]$.

I then invert this matrix, to get the Cramer-Rao lower bound; picking out the upper left component if I want to get an estimate of the asymptotic variance of the MLE estimator for $\mu$ for example.

My particular questions regarding my implementation are as follows:

1. Is it correct to replace $\mathbb{E} (I)$ in the CRLB formula by the matrix $I_{estimate}$ where I have evaluated the various components at the MLE estimated values?

2. Do I need to take account of the sample size in my ultimate estimate of the variance of the MLE estimators? For example, if we call the inverse of the information matrix, $J$, would the corresponding estimator of (asymptotic) variance just be: $J_{11}$ or should it be: $J_{11}/N$, where $N$ is the sample size?

Best,

Ben