Fisher information of a statistic I have a random sample $(X_1, X_2,...,X_n)$ and I have an estimator $\bar{X_n}=\sum_{i=1}^{n} X_i$
I need to compute the Fisher information of $\bar{X_n}$. The Fisher information is defined as $-E\left(\frac{d^2}{d\theta^2}logL\right)$, where $L$ is the likelihood function.
My question is: to compute the Fisher information of the estimator (NOT the random sample, but instead a function of the random sample), should we take the likelihood function of the random sample or the likelihood function of the distribution?
 A: There is no fisher information of the estimator, just the fisher information of a random sample $\theta$.
In Wikipedia, it says:

In mathematical statistics, the Fisher information (sometimes simply called information1) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter $\theta$ upon which the probability of X depends. 

so, it is true that fisher information is a kind of connection between two  random variables, instead of some estimator, which is a function of X.
A: I'm pretty sure that you've got some terminology mixed up. Fisher's Information is a function of the data, just like an estimator such as $\bar{X}_{n}$ that gives you an idea of how much information of the parameter of interest is contained in the sample you've acquired. You can compute Fisher's Information at an estimator (this is usually done because the F.I. depends on the unknown parameter being estimated) and we use the plug-in estimator consisting of the F.I. evaluated at the MLE (typically). 
