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How does one usually evaluate the expected value of observed Fisher information?

That is what does

$$\mathcal{I}(\theta)=-E\left[\frac{\partial^2}{\partial\theta^2}l(X,\theta)\mid\theta\right]$$

evaluate to?

Particularly, how is $E$ treated? How does $E$ apply to the derivatives?

mavavilj
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    Usually in an exercise you calculate the quantity inside the expected value (thus the derivatives of the maximum likelihood estimator) and then you use the information given (distributions of variables and estimation rules) to calculate it. – Rebellos Nov 17 '18 at 16:32
  • @Rebellos That's observed Fisher information? I'm asking expected. – mavavilj Nov 17 '18 at 16:32
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    There's a lemma which allows interchanging derivatives and $E$: https://math.stackexchange.com/a/1986477/248602 – mavavilj Nov 17 '18 at 16:34
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    If the quantity within brackets is a random variable, where is the issue with taking expectation of that quantity as usual? – StubbornAtom Nov 17 '18 at 19:17

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