Gamma distribution and Cramér-Rao bound

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.

• We're happy to help, but is this homework, or what is the status exactly? We don't usually give explicit answers for questions people have about their homework, but rather give hints so that they are able to figure it out for themselves. – gung Jul 14 '12 at 13:01
• well no, this is not a homework. we have an exam about ml estimation and I thougth about which distribution they could come up with, we did not have gamma in our excercises, so thats why I think they will take it in the exam.... so I did it on my own, I just did a mistake and I can't find it? If you see it, it would be great if you show it to me, thanks. – StochastikerUBERL Jul 14 '12 at 13:46
• Hints: 1) Look at the line of math after "if the Cramer Rao bound is reached, so:". What happened to that $\Sigma$ when you moved to the next line? 2) Look carefully at that next line. Are those expressions really equal? – jbowman Jul 14 '12 at 13:56
• ok, thanks a lot, I forgot an "N" and I did a mistake with the frac, but it is still not correct, because I get as the variance: $N \alpha \beta^2$ and this is not equal to $\beta^2 /(N \alpha)$ Mh – StochastikerUBERL Jul 14 '12 at 14:35
• First, do you know the distribution of $S = \sum_i X_i$. If so, can you find $\mathbb E(1/S)$ (not $1/\mathbb E S)$? How about $\mathbb E(1/S^2)$? This answer on math.SE may also be of some use if you get stuck. – cardinal Jul 16 '12 at 10:04