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I'm currently trying to understand more about the properties of the maximum likelihood estimator. It's known that, in the large data-limit, the MLE becomes an unbiased estimator with almost Gaussian distribution with minimum variance (it's efficient). In math, if we note $\theta_0$ the true value generating the data and $\hat \theta$ the MLE (and $I_0$ the Fisher information of the data-generating process):

$$ \sqrt n (\hat \theta - \theta_0) \rightarrow N(0, (n I_0)^{-1} ) $$

The conventional proof relies on computing the distribution of the empirical likelihood function, but are there other possibilities ? In particular, I'm wondering if somebody as come up with an explanation using information theory ideas.

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