There are two definitions of the GAMMA distribution:
I did the ML estimation, generated the Fisher Information, compared it to the Variance and the Cramer Lower Bound was reached, so the estimator is efficient. But now I tried it with this alternative:
$$g(x;\alpha,\beta) = x^{\alpha-1}\frac{\beta^{\alpha}e^{-\beta x}}{\Gamma (\alpha)}$$
I did the ML estimation and got an estimator for the $\beta$:
$$\beta_{ML}=\frac{N\alpha}{\sum x_i}$$
which is correct, as I looked up in this book: Yudi Pawitan In All Likelihood: Statistical Modelling and Inference Using Likelihood (pp. 60-61). The Fisher Information Matrix entry for $\beta$ is $=\frac{N\alpha}{\beta^2}$. Now I wanted to see if the Cramer Rao Bound is reached, so:
$$V(\beta_{ML})=V(\frac{N\alpha}{\sum x_i})=N^2\alpha^2*\frac{1}{\sum V(x_i)}$$
with $V(x_i)=\frac{\alpha}{\beta^2}$. I get the following if I insert it:
$$=\frac{N^2\alpha^2}{\alpha / \beta^2}=\frac{N^2\alpha}{\beta^2}=V(\beta_{ML})$$
Cramer-Rao-Bound:
$$V(\beta_{ML})=I^{-1}(\beta_{ML})$$
this gives:
$$\frac{N^2\alpha}{\beta^2}=(\frac{N\alpha}{\beta^2})^{-1}$$
which is not true, because:
$$\frac{N^2\alpha}{\beta^2}>\frac{\beta^2}{N\alpha}$$
So in this case,the CR lower bound is not reached, but as I said above, with the other definition of the Gamma distribution it worked. So, I made a mistake, but I can't see it.
