I have recently been studying some classical inference and I have come across this question about an efficient estimator and CRLB in a mathematical statistics textbook.
$$f(x)=\begin{cases} \frac{\lambda^{3}x^{2}}{2}e^{-\lambda x} & x>0\\ 0 & otherwise \end{cases}$$
Two things I want to calculate is
a) efficient estimator of $\lambda^{-1}$
b) Cramer-Rao Lower Bound (CRLB) for $\lambda^{-1}$
The pdf above isn't named but it looks similar to a gamma pdf in some ways.
So far I have worked out the log likelihood to be, $log(f|\lambda)=3log\lambda+2logx-\lambda x-log2$
also $\frac{\partial}{\partial\lambda}=\frac{3}{\lambda}-x$. Now i'm confused to exactly what the efficient estimator is here.
For CRLB, I have calculated $$\frac{\partial^{2}}{\partial\lambda^{2}}=-\frac{3}{\lambda^{2}}$$ $$-\mathbb{E}[-\frac{3}{\lambda^{2}}]=\frac{3}{\lambda^{2}}$$ There is no x, hence I can't take expectation of anything, is this correct?
One of things putting me off in this question is also the $\lambda^{-1}$ mentioned in the question, most of the time, I've done questions where is refers to parameter itself, not the inverse of it.