I've managed to get through a) and b) but now I'm stumped with this delta method and cramer theorem stuff:
let X be Gamma random variable with parameter $\alpha$=4 and $\lambda$=$\theta$:
a)Find the Fisher Information
b)Determine the MLE of $\theta$ an a sample from Gamma
c)Use the delta method to derive the asymptotic distribution from Gamma (at rate $\sqrt n$ or the MLE $\theta$)
d)Use Cramers's theorem to derive the asymptotic distribution at rate $\sqrt n$ or the MLE $\theta$)
my answer so far are:
a) $\alpha/\lambda^2$
b) $\hat \lambda = \alpha / \bar x$
I don't really understand the delta method formula and how to use in with the MLE that I derived.
$\sqrt{n}(X_n – \mu) \Rightarrow_n^\infty X \sim N(o,\sigma^2)$
some pointers on that would be really helpfully.
Best Answer
Your answer to b is wrong, the $n$ shouldn't be there. For c: By the Central Limit Theorem and some assumptions you can deduce a distribution for $\bar{x}$. Then use the delta method on $\bar{x}$ with the function $g(y) = 4/y$.