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Asking for a "friend" (okay, I'll be honest, asking for myself). Currently working on an assignment maximizing likelihoods as part of Monte Carlo simulations to compare estimators and everything is good until...

Compute the observed Fisher information at the MLE. Do you know another approximation to the asymptotic variance of the MLE?

I was pretty certain that the Fisher information was the only way to work out the asymptotic variance? The only other ways I can think of is calculating from the method of moments from the sample and using that to calculate the variance? Or to use multiple simulations to calculate the variance?

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The Do you know part makes it a bit hard to guess.

Some other approximations:

  1. The expected Fisher information (commonly used in generalised linear models, where it's equal to the observed Fisher information under the canonical link but not in general)
  2. The variance of the influence functions (either the empirical influence functions, giving the jackknife, or the analytic influence functions)
  3. The bootstrap
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