I have $X_1,...,X_n$ that are iid random variables with density function $$f(x) = {\alpha}x^{\alpha-1}$$
where $0<x<1$ and ${\alpha}$ is an unknown parameter.
The MLE estimator I got was $\frac{-n}{ln(n)+ln(\overline{X})}$. How do I find this MLE's asymptotic variance? I am especially stuck because of the ln function with the X bar.
The asymptotic variance of the MLE is given by $I(\alpha)^{-1}$ where $I(\alpha)$ is the Fisher information, namely, $$ I(\alpha)=-nE\left(\frac{d^2}{d\alpha^2}\log f(X;\alpha)\right)=-nE(-\alpha^{-2})=\frac{n}{\alpha^2} $$ It follows that the asymptiotic variance is $$ \frac{\alpha^2}{n}. $$