Information matrix for MLE

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Joined: Wed Mar 12, 2014 6:05 pm

Information matrix for MLE

Postby giancarp » Mon Apr 07, 2014 8:02 pm

I am computing the information matrix computing the Hessian of the loglikelihood function at the maximum likelihood estimates numerically.

In most cases I end up with meaningful information matrices, but when I have a small sample, the loglikelihood function is quite unstable and the second derivative computed numerically isn't usable.

My attention was caught by 203: "But sometimes even the second derivatives may be impossible to compute, in which case we can use an alternative form for the information matrix that requires only first derivatives of the log likelihood: I(theta) =E[score x score']"

Where can I find references about this? I am a bit confused since I am using my MLE and being a maximum I would expect that all the first derivatives should be zeros (and all the second derivatives should be negative - if one is positive then I've a saddle point). If all the first derivatives are zeros, the information matrix should be the same (and in fact this is what happens numerically).

What am I missing?

thank you very much for any suggestion or references,

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