What is the CLRB of sigma^2?

Question

I was doing this exercise and then I checked the solution, but I got the solution wrong.

$X_{1},...X_{n} \sim^{iid} N(0,\theta)$, i.e.

$f_{X_{i}} (x)= \frac{1}{\sqrt{2 \pi \theta}} -e^\frac{x^2}{2 \theta}$

b) find the CRLB for the class of unbiased estimators of $ \theta$

So I compute the Likelihood and the logarithm, then I take first and second derivative and the calculation then should be:

$ I =-E(\frac{1}{2\theta^2} - \frac{x^2}{\theta^3}$) and then CRLB = $\frac{1}{nI}$

However, doing this I came to the wrong result, can help me understand why?

Answer 1

There is no problem in your derivation, however you can have a closed-form result, which isn't in your answer, and probably that is why you got your points deducted.

You can further develop

$$ I(\theta)=-\mathbb{E}(l^{''}\left(X\vert\theta\right))=-\mathbb{E}\left(\frac{1}{2\theta^2}-\frac{X^2}{\theta^3}\right)=\frac{1}{2\theta^2} $$

and therefore the CRLB is $$ CRLB=\frac{2\theta^2}{n} $$